#6358443

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358426
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	C₄²	C₂⁵
prime radical	D₄×#135	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1777 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265639);