#417412

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

976 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)(26,30)(28,36)(31,37)(32,38)(33,40)(35,39);(9,10)(11,14)(12,13)(15,16)(17,19,20,23)(18,21,22,24)(25,26,28,30,29,32,36,38)(27,33,35,40,34,31,39,37);(1,3,4,8)(2,5,6,7)(10,13)(14,16)(19,23)(21,24)(25,27)(26,31)(28,35)(29,34)(30,37)(32,33)(36,39)(38,40);(9,11,12,15)(10,14,13,16)(18,22)(21,24)(27,34)(31,33)(35,39)(37,40)

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=f,d^2=ghi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^c=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, 324608);