#499172

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×#508	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1062 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)(11,24)(12,30)(13,19)(14,21)(15,20)(16,23)(17,27)(18,28)(22,25)(26,36)(29,39)(31,37)(32,35)(33,40)(34,38);(5,6)(7,8)(9,11)(10,17)(12,14)(13,26)(15,27)(16,29)(18,38)(19,35)(20,36)(21,24)(22,30)(23,37)(25,28)(31,34)(32,33)(39,40);(2,4)(6,8)(9,12,14,11,25,27,34,26,16,32,33,29,13,31,15,28)(10,18,20,37,19,39,40,35,23,36,38,17,22,24,21,30);(9,13,16,25)(10,19,23,22)(11,26,29,28)(12,27,32,31)(14,15,33,34)(17,35,37,30)(18,24,36,39)(20,40,38,21)

Polycyclic presentation:

2^9|c^2=e,d^2=i,e^2=hi,f^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,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,d^c=di

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, 406368);