#502103

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

794 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)(9,10,14,20,32,37,29,39,13,19,15,21,16,22,33,38)(11,23,25,18,12,28,30,40,24,17,26,34,27,35,31,36);(2,4)(5,6)(7,8)(9,11,14,25,32,12,29,30,13,24,15,26,16,27,33,31)(10,17,20,34,37,35,39,36,19,23,21,18,22,28,38,40);(6,8)(9,12,13,27)(10,18,19,34)(11,16,24,32)(14,25,15,26)(17,20,23,21)(22,40,37,36)(28,39,35,38)(29,31,33,30);(2,4)(10,19)(17,23)(18,34)(20,21)(22,37)(28,35)(36,40)(38,39)

Polycyclic presentation:

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

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, 409299);