#413265

order	512
nilpotency class	3
isoclinism	isoclinic class of #398436
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1072 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=f,d^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=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, 320461);