#218259

order	512
nilpotency class	3
isoclinism	isoclinic class of #32706
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

492 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)(11,17)(12,14)(13,21)(15,19)(16,20)(18,24)(22,23)(25,26,30,33,31,28,29,38)(27,32,36,39,37,34,35,40);(1,2)(3,6)(4,5)(7,8)(10,14)(15,23)(16,21)(17,24)(25,27,30,36,31,37,29,35)(26,32,33,39,28,34,38,40);(2,5)(6,8)(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17)(25,28,31,26)(27,34,37,32)(29,33,30,38)(35,39,36,40);(1,3,4,7)(2,6,5,8)(10,14)(15,23)(16,21)(17,24)(25,29,31,30)(26,33,28,38)(27,35,37,36)(32,39,34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cehi,d^c=di,f^c=fh

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, 125455);