#72342332

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

49471 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^10|a^2=i,d^2=j,f^2=hj,c^a=ci,d^a=dh,d^c=dj,e^b=ehj,e^d=ej,f^b=fhj,f^c=fhj,f^d=fj,g^a=gj,g^c=gj,g^e=gij,g^f=gij

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342349
maximal normal subgroup of	#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, 1, F.9);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
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.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);