#72228234

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

451793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 8 generators on 96 points:

(5,16)(10,19)(13,24)(15,26)(21,28)(22,29)(25,31)(30,32)(34,39)(35,43)(37,48)(41,50)(44,55)(47,58)(53,60)(62,64)(66,71)(69,80)(72,81)(73,82)(77,88)(79,90)(84,91)(89,95);(4,14)(9,18)(12,23)(15,26)(20,27)(22,29)(25,31)(30,32)(34,39)(36,46)(40,49)(42,51)(44,55)(47,58)(53,60)(57,63)(68,78)(69,80)(73,82)(74,83)(76,87)(77,88)(84,91)(85,92);(3,11)(8,17)(12,23)(13,24)(20,27)(21,28)(25,31)(30,32)(35,43)(37,48)(40,49)(42,51)(44,55)(47,58)(52,59)(54,61)(67,75)(68,78)(72,81)(73,82)(77,88)(79,90)(85,92)(86,93);(2,7)(5,16)(8,17)(9,18)(13,24)(15,26)(20,27)(25,31)(34,39)(36,46)(37,48)(40,49)(44,55)(45,56)(54,61)(62,64)(66,71)(67,75)(69,80)(73,82)(76,87)(79,90)(85,92)(94,96);(1,2)(3,8)(4,9)(5,10)(6,7)(11,17)(12,20)(13,21)(14,18)(15,22)(16,19)(23,27)(24,28)(25,30)(26,29)(31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,49)(44,52)(45,53)(46,50)(47,54)(48,51)(55,59)(56,60)(57,62)(58,61)(63,64)(65,66)(67,72)(68,73)(69,74)(70,71)(75,81)(76,84)(77,85)(78,82)(79,86)(80,83)(87,91)(88,92)(89,94)(90,93)(95,96);(1,3)(2,8)(4,12)(5,13)(6,11)(7,17)(9,20)(10,21)(14,23)(15,25)(16,24)(18,27)(19,28)(22,30)(26,31)(29,32)(33,35)(34,40)(36,44)(37,45)(38,43)(39,49)(41,52)(42,53)(46,55)(47,57)(48,56)(50,59)(51,60)(54,62)(58,63)(61,64)(65,67)(66,72)(68,76)(69,77)(70,75)(71,81)(73,84)(74,85)(78,87)(79,89)(80,88)(82,91)(83,92)(86,94)(90,95)(93,96);(1,4)(2,9)(3,12)(5,15)(6,14)(7,18)(8,20)(10,22)(11,23)(13,25)(16,26)(17,27)(19,29)(21,30)(24,31)(28,32)(33,36)(34,41)(35,44)(37,47)(38,46)(39,50)(40,52)(42,54)(43,55)(45,57)(48,58)(49,59)(51,61)(53,62)(56,63)(60,64)(65,68)(66,73)(67,76)(69,79)(70,78)(71,82)(72,84)(74,86)(75,87)(77,89)(80,90)(81,91)(83,93)(85,94)(88,95)(92,96);(1,5)(2,10)(3,13)(4,15)(6,16)(7,19)(8,21)(9,22)(11,24)(12,25)(14,26)(17,28)(18,29)(20,30)(23,31)(27,32)(33,37)(34,42)(35,45)(36,47)(38,48)(39,51)(40,53)(41,54)(43,56)(44,57)(46,58)(49,60)(50,61)(52,62)(55,63)(59,64)(65,69)(66,74)(67,77)(68,79)(70,80)(71,83)(72,85)(73,86)(75,88)(76,89)(78,90)(81,92)(82,93)(84,94)(87,95)(91,96)

Polycyclic presentation:

2^11|e^a=ei,e^b=eij,e^d=ek,f^a=fij,f^c=fk,f^d=fj,g^b=gk,g^c=gj,g^d=gij,h^a=hk,h^b=hj,h^c=hij,h^d=hk

Conjugacy classes & character table

263 conjugacy classes — view the full character table.

Relationships

maximal quotient of

H(3,𝔽₁₆)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);