#495994

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×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×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

552 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)(9,10,14,18,15,20,30,19)(11,21,23,32,25,38,24,40)(12,17,26,35,27,36,13,28)(16,31,33,39,34,29,37,22);(1,3,5,4)(2,6,7,8);(9,11)(10,16)(12,22)(13,29)(14,23)(15,25)(17,32)(18,33)(19,37)(20,34)(21,28)(24,30)(26,31)(27,39)(35,38)(36,40);(1,4,5,3)(2,6,7,8)(9,12,14,26,15,27,30,13)(10,17,18,35,20,36,19,28)(11,22,23,31,25,39,24,29)(16,32,33,38,34,40,37,21)

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,f^c=fh,h^c=hi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403190);