#72249529

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 39948 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:

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

Polycyclic presentation:

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