#72341056

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

42899 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|b^2=i,c^2=ij,e^2=ij,f^2=i,b^a=bj,c^a=ci,e^a=eij,e^b=eij,e^c=eh,e^d=eij,f^a=fi,f^b=fhij,f^c=fhj,f^d=fi,g^b=gij,g^c=gj

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341060, #72341176, #72341187
maximal normal subgroup of	#72341060, #72341176, #72341187

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