#387564

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

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

818 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(25,26,30,31,29,33,27,34)(28,32,37,38,36,39,35,40);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,13,14,15)(17,18)(19,22)(20,21)(23,24)(25,27,29,30)(26,31,33,34)(28,35,36,37)(32,38,39,40);(1,4,5,8)(2,6,3,7)(10,14)(13,15)(18,21)(22,24)(25,28,29,36)(26,32,33,39)(27,35,30,37)(31,38,34,40);(2,3)(6,7)(11,16)(13,15)(17,19,20,23)(18,22,21,24)(28,36)(32,39)(35,37)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ef,d^2=g,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, 294760);