#72249491

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 10870 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:

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

Polycyclic presentation:

2^10|a^2=i,b^2=i,f^2=j,b^a=bi,e^a=eh,e^c=ehj,e^d=ei,f^a=fi,f^b=fhj,f^c=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, #72256271
maximal normal subgroup of	#72249567, #72256271

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, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*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, 3, 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);