#387243

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

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

890 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=fg,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=dei,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294439);