#494289

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	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1170 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)(3,4)(5,6,8,7)(9,10)(11,20)(12,16)(13,28)(14,17)(15,22)(18,25)(19,26)(21,36)(23,38)(24,35)(27,32)(29,34)(30,39)(31,37)(33,40);(2,4)(5,7,8,6)(9,11)(10,15)(12,24)(13,29)(14,21)(16,20)(17,31)(18,34)(19,35)(22,25)(23,26)(27,39)(28,30)(32,36)(33,38)(37,40);(10,16)(12,25)(13,26)(15,20)(17,32)(18,19)(22,24)(23,29)(27,40)(28,33)(30,38)(31,36)(34,35)(37,39);(10,17)(11,21)(16,32)(18,33)(19,28)(22,37)(23,29)(24,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 401485);