#72249489

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 20862 conjugacy classes of subgroups; only the 29393 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 7 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=i,f^2=j,b^a=bi,e^c=eh,e^d=ei,f^a=fij,f^b=fh,f^c=fj,f^e=fhj,g^b=gi,g^e=gj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249567
maximal normal subgroup of	#72249567

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