#410539

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2537
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

813 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,25)(13,19)(14,17)(15,20)(18,28)(21,30)(22,36)(23,31)(24,32)(26,33)(27,34)(29,35)(37,39)(38,40);(9,11,15,21,13,24,14,23)(10,16,17,31,19,32,20,30)(12,22,28,35,26,38,27,37)(18,29,25,36,34,39,33,40);(1,3,5,4)(2,6,7,8)(10,17,19,20)(11,21,24,23)(12,26)(18,33,34,25)(22,37,38,35)(27,28)(29,39)(36,40);(1,4,5,3)(2,6,7,8)(9,12,14,27,13,26,15,28)(10,18,20,33,19,34,17,25)(11,22,23,37,24,38,21,35)(16,29,30,40,32,39,31,36)

Polycyclic presentation:

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

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, 317735);