#398281

order	512
nilpotency class	3
isoclinism	isoclinic class of #394353
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1131 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=f,d^2=e,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,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, 305477);