#72318787

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 10770 conjugacy classes of subgroups; only the 29605 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:

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

Polycyclic presentation:

2^10|a^2=h,f^2=j,b^a=bi,c^a=ch,c^b=ci,d^c=dhi,e^c=ej,f^a=fi,f^b=fj,f^c=fj,g^a=gj,g^b=gi,g^c=gj,g^f=gj

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318803, #72318828
maximal normal subgroup of	#72318803, #72318828

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