#478453

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efgi,b^2=efi,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,d^c=de,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385649);