#409934

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

904 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=fh,c^2=fi,d^2=efi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 317130);