#502876

order	512
nilpotency class	4
isoclinism	isoclinic class of #501072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

909 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 410072);