#502497

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

706 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=efh,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, 409693);