#500840

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

994 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)(9,10,15,20)(11,21,24,39)(12,25,28,38)(13,29,26,40)(14,31,30,19)(16,32,35,23)(17,36,34,27)(18,37,33,22);(5,6)(7,8)(9,11,22,27,14,12,26,32,15,24,37,36,30,28,13,23)(10,16,29,38,19,34,33,39,20,35,40,25,31,17,18,21);(2,4)(6,8);(9,12,22,32,14,24,26,36,15,28,37,23,30,11,13,27)(10,17,29,21,19,16,33,38,20,34,40,39,31,35,18,25)

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

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, 408036);