#504083

order	512
nilpotency class	4
isoclinism	isoclinic class of #503866
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂²×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

437 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411279);