#7501578

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500736
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
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1900 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,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, 7408774);