#500842

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

991 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408038);