#72341237

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

45573 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^10|d^2=hj,g^2=hj,c^b=ci,d^a=dj,d^b=dj,d^c=dh,e^d=eij,f^b=fj,f^d=fhij,g^a=gi,g^b=gi,g^c=ghi,g^d=ghj,g^e=gj,g^f=gj

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341243
maximal normal subgroup of	#72341243

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