#500660

order	512
nilpotency class	4
isoclinism	isoclinic class of #499891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3147	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 407856);