#417010

order	512
nilpotency class	3
isoclinism	isoclinic class of #394464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

910 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)(3,7)(4,5)(6,8)(9,10,14,20,15,18,13,19)(11,21,23,35,24,16,29,32)(12,25,28,34,27,36,30,17)(22,33,26,37,38,40,39,31);(9,11,13,23,15,24,14,29)(10,16,19,32,18,21,20,35)(12,26,28,22,27,39,30,38)(17,33,25,31,34,40,36,37);(2,5)(6,7)(9,12)(10,17,18,34)(11,22)(13,28)(14,30)(15,27)(16,31,21,37)(19,25,20,36)(23,39)(24,38)(26,29)(32,40,35,33);(1,3,4,8)(2,6,5,7)(12,27)(17,34)(22,38)(25,36)(26,39)(28,30)(31,37)(33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=e,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,d^c=di,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, 324206);