#499173

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1118 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)(9,10)(11,21)(12,16)(13,19)(14,29)(15,18)(17,22)(20,26)(23,32)(24,34)(25,35)(27,30)(28,31)(33,38)(36,39)(37,40);(5,6)(7,8)(10,16)(11,22)(12,13)(14,26)(17,32)(18,30)(19,20)(23,24)(25,38)(27,28)(29,31)(33,40)(34,36)(37,39);(2,4)(6,8)(9,11,12,23,26,39,28,38,15,25,27,37,14,24,13,22)(10,17,19,34,29,40,30,35,18,33,31,36,20,32,16,21);(10,18)(16,30)(17,33)(19,31)(20,29)(21,35)(32,40)(34,36)

Polycyclic presentation:

2^9|c^2=e,e^2=hi,f^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,e^b=eh,f^b=fhi,g^b=ghi,h^b=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, 406369);