#72340990

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

43107 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)(11,15)(12,17)(13,19)(14,16)(18,24)(20,23)(21,22)(25,26)(27,32)(28,33)(29,30)(31,34)(35,37)(36,39)(38,40);(9,11)(10,15)(12,19)(13,17)(14,21)(16,22)(18,23)(20,24)(25,27)(26,32)(28,35)(29,36)(30,39)(31,38)(33,37)(34,40);(9,11)(10,15)(12,18)(13,20)(14,21)(16,22)(17,24)(19,23)(25,27)(26,32)(28,36)(29,35)(30,37)(31,38)(33,39)(34,40);(3,7)(6,8)(10,16)(13,24)(15,22)(17,20)(25,28,31,29)(26,33,34,30)(27,35,38,36)(32,37,40,39);(2,5)(3,7)(9,12,14,23)(10,17,16,20)(11,19,21,18)(13,22,24,15)(25,29,31,28)(26,33,34,30)(27,36,38,35)(32,37,40,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,20)(14,21)(16,22)(17,24)(19,23)(25,30,31,33)(26,28,34,29)(27,37,38,39)(32,36,40,35);(1,3)(2,6)(4,7)(5,8)(9,13,14,24)(10,18,16,19)(11,20,21,17)(12,22,23,15)(25,27)(26,32)(28,36)(29,35)(30,37)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^10|d^2=j,e^2=ij,f^2=j,g^2=i,d^a=di,d^c=dj,e^a=ej,e^c=eij,f^a=fj,f^b=fj,f^d=fj,f^e=fh,g^a=gi,g^b=gi,g^d=ghj,g^e=ghij

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72340993
maximal subgroup of	#72340993, #72341145
maximal normal subgroup of	#72340993, #72341145

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