#6358595

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358586
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	C₂×#547	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1733 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)(11,19)(12,16)(13,27)(14,17)(15,22)(18,28)(20,35)(21,31)(23,30)(24,38)(25,39)(26,33)(29,37)(32,36)(34,40);(9,11,14,23)(10,15,17,31)(12,24,26,20)(13,21,28,22)(16,32,33,29)(18,30,27,19)(25,37,40,36)(34,35,39,38);(2,5)(3,8)(6,7)(9,12,14,26)(10,16,17,33)(11,20,23,24)(13,25,28,40)(15,29,31,32)(18,34,27,39)(19,35,30,38)(21,36,22,37);(11,21)(13,28)(15,30)(18,27)(19,31)(20,36)(22,23)(24,37)(25,40)(29,38)(32,35)(34,39);(10,17)(15,31)(16,33)(18,27)(19,30)(29,32)(34,39)(35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265791);