#495995

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

546 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)(3,8)(4,6)(5,7)(9,10,14,18,15,20,31,19)(11,21,24,36,26,40,25,33)(12,17,27,37,28,38,13,29)(16,32,34,22,35,30,39,23);(1,3,5,4)(2,6,7,8);(9,11)(10,16)(12,23)(13,30)(14,24)(15,26)(17,36)(18,34)(19,39)(20,35)(21,29)(22,28)(25,31)(27,32)(33,38)(37,40);(1,4,5,3)(2,6,7,8)(9,12,14,27,15,28,31,13)(10,17,18,37,20,38,19,29)(11,22,24,30,26,23,25,32)(16,33,34,21,35,36,39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=eg,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=di,f^c=fh,h^c=hi

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, 403191);