#72265952

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 10421 conjugacy classes of subgroups; only the 29361 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:

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

Polycyclic presentation:

2^10|a^2=j,c^2=j,d^c=dh,e^b=eh,f^b=fi,f^d=fj,g^a=gi,g^e=gj

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265968
maximal normal subgroup of	#72265968
maximal quotient of	#72266583

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