#6912214

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1599 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308682, #72309006
maximal normal subgroup of	#72308682, #72309006

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6819410);