#72249496

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 38838 conjugacy classes of subgroups; only the 29385 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:

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

Polycyclic presentation:

2^10|a^2=i,b^2=i,f^2=j,b^a=bi,c^a=ci,d^a=di,e^a=eh,e^c=ej,e^d=ehi,f^a=fi,f^b=fhj,f^d=fj,f^e=fh,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, 1, F.9);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
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);