#415477

order	512
nilpotency class	3
isoclinism	isoclinic class of #401626
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

895 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)(9,10,14,18,15,20,13,19)(11,21,24,32,25,36,23,16)(12,26,29,17,30,34,28,35)(22,31,27,33,39,40,38,37);(9,11,13,23,15,25,14,24)(10,16,18,21,20,32,19,36)(12,27,28,22,30,38,29,39)(17,33,34,40,35,37,26,31);(9,12,15,30)(10,17,20,35)(11,22,25,39)(13,29,14,28)(16,31,32,40)(18,26,19,34)(21,37,36,33)(23,27,24,38);(1,3,4,8)(2,5,6,7)

Polycyclic presentation:

2^9|a^2=gh,b^2=e,c^2=i,d^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,e^c=ei,f^c=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, 322673);