#502868

order	512
nilpotency class	4
isoclinism	isoclinic class of #501076
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

857 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)(3,7)(4,5)(6,8)(9,10)(11,20)(12,26)(13,17)(14,30)(15,18)(16,22)(19,32)(21,23)(24,34)(25,33)(27,39)(28,40)(29,31)(35,36)(37,38);(2,5)(6,7)(9,11)(10,16)(12,27)(13,22)(14,23)(15,25)(17,20)(18,34)(19,33)(21,29)(24,32)(26,36)(28,35)(30,37)(31,38)(39,40);(9,12,26,10,15,30,31,19,13,28,40,17,32,29,14,18)(11,21,36,34,25,27,38,16,22,37,39,33,24,35,23,20);(1,3,4,8)(2,6,5,7)(10,17)(12,28)(16,20)(18,19)(21,37)(27,35)(29,30)(33,34)

Polycyclic presentation:

2^9|c^2=fh,d^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410064);