#72323162

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#12067
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	#1311	C₂⁴
prime radical	#513080	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ #122214 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

747 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 4 generators on 96 points:

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

Polycyclic presentation:

2^11|a^2=e,c^2=eg,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,b^a=bf,c^b=chjk,d^a=di,d^b=dik,d^c=dk,e^b=ejk,f^a=fh,f^b=fhjk,f^c=fj,f^e=fk,g^b=gk,h^a=hj,h^b=hjk,h^c=hk,j^a=jk,j^b=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323164
maximal normal subgroup of	#72323164

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