#72340424

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 6518 conjugacy classes of subgroups; only the 29389 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 32 points:

(11,15)(13,16)(18,23)(20,24)(21,29)(27,30);(17,18)(19,24)(20,25)(21,26)(22,23)(27,32)(28,30)(29,31);(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,16,15,13);(3,7)(5,8)(11,15)(13,16)(19,25)(20,24)(27,30)(28,32);(1,3)(2,5)(4,7)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,25)(18,24,23,20)(21,30,29,27)(26,28,31,32);(9,10)(11,13)(12,14)(15,16)(17,20,22,24)(18,25,23,19)(21,28,29,32)(26,30,31,27);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,26)(19,27)(20,28)(22,29)(23,31)(24,32)(25,30)

Polycyclic presentation:

2^10|c^2=i,e^2=hj,f^2=j,b^a=bj,e^a=eh,e^c=eh,e^d=eij,f^a=fj,f^c=fh,f^d=fj,g^a=gj,g^c=gi,g^e=gj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340428, #72340445
maximal normal subgroup of	#72340428, #72340445

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