#7504046

order	512
nilpotency class	3
isoclinism	isoclinic class of #7501009
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411242);