#503103

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1325 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)(14,29)(15,16)(17,26)(19,24)(21,35)(22,34)(25,38)(27,37)(28,31)(30,39)(32,40)(33,36);(1,2,4,3)(5,6)(7,8)(9,11,13,15)(10,16,18,20)(12,21,25,27)(14,22,28,24)(17,30,32,33)(19,31,34,29)(23,37,38,35)(26,36,40,39);(6,8)(9,12,24,39,15,27,28,40,13,25,22,36,11,21,14,26)(10,17,29,35,20,33,34,38,18,32,31,37,16,30,19,23);(1,3,4,2)(10,18)(16,20)(17,32)(19,34)(23,38)(29,31)(30,33)(35,37)

Polycyclic presentation:

2^9|b^2=e,c^2=fh,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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410299);