#494284

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
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 ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1094 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)(11,20)(12,15)(13,28)(14,17)(16,21)(18,24)(19,26)(22,39)(23,36)(25,35)(27,30)(29,34)(31,37)(32,38)(33,40);(2,4)(5,7,8,6)(10,15)(12,24)(13,26)(16,20)(17,30)(18,19)(21,25)(22,29)(27,37)(28,31)(32,36)(33,39)(34,35)(38,40);(9,11,14,23)(10,16,17,32)(12,25,27,40)(13,29,26,22)(15,20,30,36)(18,34,31,39)(19,35,28,33)(21,37,38,24);(10,17)(15,30)(16,32)(18,31)(19,28)(20,36)(33,35)(34,39)

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,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, 401480);