#72333433

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 7043 conjugacy classes of subgroups; only the 29289 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 7 generators on 40 points:

(3,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,19)(20,22)(21,24)(29,37)(33,38)(34,39)(36,40);(10,14)(15,18)(16,22)(23,24)(25,26)(27,28)(29,38)(30,31)(32,35)(33,37)(34,40)(36,39);(2,5)(3,7)(26,31)(28,35)(29,37)(34,39);(12,20)(16,22)(17,21)(23,24)(29,37)(33,38)(34,39)(36,40);(9,11)(10,15)(12,21)(13,19)(14,18)(16,24)(17,20)(22,23)(25,27)(26,28)(29,34)(30,32)(31,35)(33,36)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,17)(13,20)(14,22)(15,23)(18,24)(19,21)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,38)(37,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,18)(12,16)(13,14)(15,19)(17,24)(20,22)(21,23)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)

Polycyclic presentation:

2^10|b^a=bj,c^b=ch,f^b=fh,f^c=fi,f^d=fj,f^e=fhj,g^a=gi,g^b=ghj,g^c=gi,g^d=gh,g^e=gj

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72333496, #72334158

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