#72322532

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#367
commutator	C₂×C₁₆	C₂⁴×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁵
prime radical	#3444	D₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #60204 ◁ #7622657 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2896 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 80 points:

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

Polycyclic presentation:

2^11|a^2=f,e^2=gijk,f^2=h,g^2=ijk,i^2=jk,j^2=k,c^b=chk,d^b=dk,e^a=eg,f^e=fk,g^a=gi,i^a=ij,j^a=jk

Conjugacy classes & character table

236 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322589, #72323095, #72323105
maximal normal subgroup of	#72322589, #72323095, #72323105

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> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 5, F.7*F.9*F.10*F.11);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10*F.11);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);