#501007

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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	C₂²×#509	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1083 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)(3,4)(5,6,7,8)(9,10)(11,16)(12,18)(13,17)(14,19)(15,20)(21,35)(22,30)(23,31)(24,32)(25,34)(26,33)(27,36)(28,37)(29,39)(38,40);(2,4)(9,11)(10,16)(12,22)(13,23)(14,21)(15,28)(17,31)(18,30)(19,35)(20,37)(24,27)(25,38)(26,39)(29,33)(32,36)(34,40);(6,8)(10,17)(11,21)(12,15)(14,27)(16,29)(19,34)(22,24)(23,39)(25,26)(28,38)(30,40)(31,35)(32,37)(33,36);(2,4)(9,12,13,15)(10,18,17,20)(11,22,23,28)(14,25,26,27)(16,30,31,37)(19,34,33,36)(21,38,39,24)(29,32,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408203);