#512068

order	512
nilpotency class	4
isoclinism	isoclinic class of #511755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,h^a=hi,f^c=fh,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, 419264);