#72249498

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 39359 conjugacy classes of subgroups; only the 29387 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:

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

Polycyclic presentation:

2^10|b^2=i,b^a=bi,c^a=ci,e^c=ej,e^d=ei,f^a=fj,f^b=fh,g^a=gi,g^b=gi,g^e=gh,g^f=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> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);