#503144

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

1184 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)(9,10,14,18)(11,19,22,33)(12,24,26,40)(13,27,25,17)(15,28,29,20)(16,31,32,21)(23,38,36,35)(30,37,34,39);(1,2,4,3)(11,20)(12,21)(13,25)(15,19)(16,24)(17,27)(22,28)(23,39)(26,31)(29,33)(30,38)(32,40)(34,35)(36,37);(5,6,7,8)(9,11,21,37,13,23,26,28,14,22,31,39,25,36,12,20)(10,15,24,35,17,30,32,33,18,29,40,38,27,34,16,19);(1,3,4,2)(11,22)(15,29)(19,33)(20,28)(23,36)(30,34)(35,38)(37,39)

Polycyclic presentation:

2^9|a^2=i,b^2=e,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, 410340);