#501001

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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×#508	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

1040 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)(5,6,7,8)(11,20)(16,31)(21,37)(22,30)(23,32)(27,40)(33,34)(36,38);(2,4)(9,10,13,18)(11,21,20,37)(12,17,15,29)(14,26,25,39)(16,32,31,23)(19,24,35,28)(22,38,30,36)(27,33,40,34);(6,8)(9,11)(10,16)(12,22)(13,20)(14,27)(15,30)(17,33)(18,31)(19,36)(21,26)(23,28)(24,32)(25,40)(29,34)(35,38)(37,39);(2,4)(9,12,13,15)(10,17,18,29)(11,22,20,30)(14,24,25,28)(16,33,31,34)(19,26,35,39)(21,38,37,36)(23,27,32,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,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,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408197);