#494290

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1062 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)(11,22)(12,25)(13,29)(14,34)(15,38)(16,18)(17,24)(19,33)(20,37)(21,32)(23,27)(26,31)(28,30)(35,40)(36,39);(2,4)(5,7,8,6)(9,11)(10,17)(12,18)(13,30)(14,35)(15,39)(16,23)(19,36)(20,31)(21,28)(22,29)(24,33)(25,37)(26,32)(27,34)(38,40);(9,12,16,17)(10,11,18,23)(13,31,14,36)(15,40,32,28)(19,30,20,35)(21,39,38,26)(22,33,27,37)(24,34,25,29);(10,18)(11,23)(19,20)(21,38)(24,25)(26,39)(29,34)(30,35)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,h^a=hi,d^b=di,e^b=eh,f^b=fh,h^b=hi,e^c=eh,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, 401486);