#415366

order	512
nilpotency class	3
isoclinism	isoclinic class of #394464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#393
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,16)(12,24,26,35)(13,18)(14,29)(15,19)(17,27,34,28)(20,33,36,40)(21,31)(22,32)(23,38)(25,30,37,39);(9,11)(10,16)(12,25,26,37)(13,22)(14,23)(15,21)(17,33,34,40)(18,32)(19,31)(20,28,36,27)(24,39,35,30)(29,38);(9,12,13,28,15,26,14,27)(10,17,18,35,19,34,29,24)(11,20,22,25,21,36,23,37)(16,30,32,40,31,39,38,33);(1,3,4,8)(2,5,6,7)(11,21)(16,31)(20,36)(22,23)(25,37)(30,39)(32,38)(33,40)

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=ef,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=di,e^c=ei,f^c=fi,g^c=gi

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, 322562);