#72312909

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 8137 conjugacy classes of subgroups; only the 29421 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)(3,7)(4,5)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,24)(20,28)(21,29)(22,23)(25,30)(26,32)(27,31);(2,5)(3,8)(18,23)(19,25)(21,28)(26,31);(1,3,4,8)(2,6,5,7)(11,16)(12,14)(18,23)(21,28)(24,30)(27,32);(17,19)(18,24)(20,26)(21,27)(22,25)(23,30)(28,32)(29,31);(9,11)(10,14)(12,15)(13,16)(17,20)(18,21)(19,26)(22,29)(23,28)(24,27)(25,31)(30,32);(1,3)(2,6)(4,8)(5,7)(9,12,13,14)(10,11,15,16)(17,21)(18,20)(19,27)(22,28)(23,29)(24,26)(25,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,24)(20,26)(21,27)(22,25)(23,30)(28,32)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312921, #72342349
maximal normal subgroup of	#72312921, #72342349

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.8);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9*F.10);;
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.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*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);