#417362

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
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

868 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)(3,7,8,5)(17,18)(19,21)(20,22)(23,24)(27,33)(28,35)(30,36)(32,38)(34,39)(37,40);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19,20,23)(18,21,22,24)(25,26)(27,30)(28,32)(29,31)(33,36)(34,37)(35,38)(39,40);(10,13)(14,16)(19,23)(21,24)(25,27,28,33,29,34,35,39)(26,30,32,36,31,37,38,40);(9,11,12,15)(10,14,13,16)(18,22)(21,24)(26,31)(30,37)(32,38)(36,40)

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=f,d^2=gh,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324558);