#6459592

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2358 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)(3,5)(4,6)(7,8)(9,10,13,18,15,19,14,30)(11,20,24,32,22,33,23,16)(12,25,27,35,29,39,28,17)(21,31,38,36,26,34,37,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,26)(13,23)(14,24)(15,22)(17,34)(18,33)(19,32)(20,30)(21,29)(25,36)(27,38)(28,37)(31,35)(39,40);(2,6)(3,7)(9,12,13,27,15,29,14,28)(10,17,30,39,19,35,18,25)(11,21,23,37,22,26,24,38)(16,31,20,36,32,34,33,40);(3,7)(5,8)(11,22)(16,32)(20,33)(21,26)(23,24)(31,34)(36,40)(37,38);(9,13,15,14)(10,18,19,30)(11,23,22,24)(12,27,29,28)(16,33,32,20)(17,25,35,39)(21,37,26,38)(31,40,34,36)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366788);