#483161

order	512
nilpotency class	4
isoclinism	isoclinic class of #482075
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂×#466	C₄
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

783 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,e^2=hi,f^2=gh,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, 390357);