#494907

order	512
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

836 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)(3,4)(5,6,8,7)(9,10,16,23)(11,19,27,38)(12,18,30,36)(13,20,17,32)(14,22,31,33)(15,21,24,39)(25,35,28,40)(26,37,29,34);(5,7,8,6)(9,11,24,29,17,28,14,12,16,27,15,26,13,25,31,30)(10,18,33,40,20,34,21,19,23,36,22,35,32,37,39,38);(2,4)(9,12,14,25,13,29,24,27,16,30,31,28,17,26,15,11)(10,19,21,37,32,40,33,36,23,38,39,34,20,35,22,18);(9,13,16,17)(10,20,23,32)(11,25,27,28)(12,29,30,26)(14,24,31,15)(18,34,36,37)(19,35,38,40)(21,22,39,33)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,d^2=i,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,c^b=ci

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402103);