#72340970

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

45621 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:

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

Polycyclic presentation:

2^10|f^2=hj,g^2=hi,d^c=di,e^c=eij,f^a=fi,f^b=fj,f^c=fij,f^d=fh,f^e=fhj,g^a=gj,g^b=gij,g^c=gi,g^d=ghi,g^e=gij,g^f=ghij

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72340973, #72341182
maximal subgroup of	#72340973, #72341121, #72341171
maximal normal subgroup of	#72340973, #72341121, #72341171

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