#72342326

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 23223 conjugacy classes of subgroups; only the 29401 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:

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

Polycyclic presentation:

2^10|d^2=j,e^2=ij,f^2=hj,g^2=j,d^a=dh,d^c=dj,e^b=ehj,e^d=ei,f^b=fhj,f^c=fhj,f^d=fi,f^e=fij,g^a=gi,g^d=gj,g^e=gi,g^f=gij

Conjugacy classes & character table

166 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, 4, F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.10);;
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.9);;
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.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);