#416368

order	512
nilpotency class	3
isoclinism	isoclinic class of #395944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1004 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=e,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,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, 323564);