#6358440

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	C₂×#3287	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1781 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)(11,20)(12,17)(13,18)(14,27)(15,19)(16,21)(22,37)(23,30)(24,33)(25,40)(26,35)(28,32)(29,36)(31,38)(34,39);(9,11,15,23)(10,16,19,32)(12,21,14,28)(13,25,26,38)(17,30,27,20)(18,34,35,37)(22,29,39,24)(31,33,40,36);(2,5)(3,8)(6,7)(9,12,15,14)(10,17,19,27)(11,21,23,28)(13,24,26,29)(16,30,32,20)(18,33,35,36)(22,38,39,25)(31,34,40,37);(9,13,15,26)(10,18,19,35)(11,22,23,39)(12,24,14,29)(16,31,32,40)(17,33,27,36)(20,37,30,34)(21,38,28,25);(10,19)(16,32)(17,27)(18,35)(20,30)(31,40)(33,36)(34,37)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^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, 6265636);