#72318648

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 7290 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:

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

Polycyclic presentation:

2^10|e^2=j,c^a=ch,c^b=chj,d^a=di,d^b=dhi,e^a=ehj,e^b=ej,e^c=ej,e^d=ej,f^a=fi,f^b=fi,g^b=gij,g^d=gj,g^f=gj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318725
maximal normal subgroup of	#72318725

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