#494497

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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×#508	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1376 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fh,g^c=gh,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, 401693);