#72311959

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 12579 conjugacy classes of subgroups; only the 29345 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)(11,14)(12,15)(13,16)(17,18)(19,20)(21,29)(22,23)(24,32)(25,30)(26,27)(28,31);(9,11,13,15)(10,14,16,12)(17,19)(18,20)(21,25)(22,26)(23,27)(24,28)(29,30)(31,32);(17,19)(18,20)(21,25)(22,26)(23,27)(24,28)(29,30)(31,32);(9,11)(10,12)(13,15)(14,16)(18,23)(20,27)(24,29)(28,30);(2,5)(3,8)(9,12)(10,15)(11,16)(13,14)(17,20,22,27)(18,19,23,26)(21,30,32,28)(24,31,29,25);(1,3)(2,6)(4,8)(5,7)(17,21,22,32)(18,24,23,29)(19,25,26,31)(20,28,27,30);(1,2)(3,6)(4,5)(7,8)(17,19)(18,20)(21,31)(22,26)(23,27)(24,30)(25,32)(28,29)

Polycyclic presentation:

2^10|b^2=h,e^2=j,f^2=j,d^a=dhj,d^b=dh,e^a=ei,e^b=ehj,e^c=ej,e^d=ehj,f^a=fij,f^e=fij,g^e=gij,g^f=gj

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311978
maximal normal subgroup of	#72311978

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