#483122

order	512
nilpotency class	4
isoclinism	isoclinic class of #482075
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=hi,f^2=ghi,h^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,e^b=eh,f^b=fgh,h^b=hi

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, 390318);