#501066

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
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×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1071 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)(9,10,14,18,26,33,38,37,13,17,25,32,15,19,28,35)(11,12,21,23,36,29,40,39,20,16,30,31,22,24,34,27);(2,4)(5,6)(7,8)(9,11)(10,16)(12,17)(13,20)(14,21)(15,22)(18,31)(19,29)(23,32)(24,33)(25,30)(26,36)(27,37)(28,34)(35,39)(38,40);(6,8)(9,12,13,16)(10,11,17,20)(14,27,25,39)(15,29,26,24)(18,34,32,40)(19,36,33,22)(21,37,30,35)(23,38,31,28);(2,4)(10,17)(11,20)(18,32)(19,33)(21,30)(22,36)(34,40)(35,37)

Polycyclic presentation:

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

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, 408262);