#502108

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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

825 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)(9,10,15,21,12,29,31,32,13,19,14,20,16,18,30,33)(11,22,27,36,24,35,34,23,25,17,26,37,28,39,38,40);(2,4)(5,6)(7,8)(9,11,14,26,12,24,30,38,13,25,15,27,16,28,31,34)(10,17,20,36,29,39,33,23,19,22,21,37,18,35,32,40);(6,8)(9,10,13,19)(11,23,25,40)(12,18,16,29)(14,32,15,33)(17,34,22,38)(20,30,21,31)(24,36,28,37)(26,39,27,35);(2,4)(9,12,13,16)(10,18,19,29)(11,24,25,28)(14,30,15,31)(17,35,22,39)(20,32,21,33)(23,36,40,37)(26,38,27,34)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^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, 409304);