#398875

order	512
nilpotency class	3
isoclinism	isoclinic class of #395238
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

967 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)(9,10,14,20)(11,21,24,17)(12,25,27,18)(13,29,15,19)(16,30,32,23)(22,33,37,35)(26,34,28,39)(31,36,38,40);(9,11,13,23,14,24,15,30)(10,16,19,21,20,32,29,17)(12,22,26,36,27,37,28,40)(18,31,34,35,25,38,39,33);(1,3,4,8)(2,5,6,7)(9,11,14,24)(10,17,20,21)(12,22,27,37)(13,23,15,30)(16,29,32,19)(18,33,25,35)(26,36,28,40)(31,39,38,34);(9,12,13,26,14,27,15,28)(10,18,19,34,20,25,29,39)(11,22,23,36,24,37,30,40)(16,31,21,35,32,38,17,33)

Polycyclic presentation:

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

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, 306071);