#503145

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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 ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1210 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)(6,8)(9,10)(11,23)(12,27)(13,22)(14,20)(15,33)(16,19)(17,25)(18,24)(21,29)(26,40)(28,38)(30,35)(31,39)(32,37)(34,36);(1,2,4,3)(9,11)(10,17)(12,28)(13,24)(14,25)(15,31)(16,34)(18,19)(20,23)(21,27)(22,36)(26,29)(30,32)(33,35)(37,39)(38,40);(5,6,7,8)(9,12,29,25,16,30,32,34,14,26,28,11,13,31,15,24)(10,18,33,39,22,23,38,40,20,36,37,35,19,17,21,27);(1,3,4,2)(9,13,14,16)(10,19,20,22)(11,24,25,34)(12,30,26,31)(15,28,32,29)(17,18,23,36)(21,37,38,33)(27,39,40,35)

Polycyclic presentation:

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

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, 410341);