#474147

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=efi,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de

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, 381343);