#494505

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1330 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)(9,10,14,21,12,19,30,17,16,23,13,20,15,22,29,35)(11,24,27,40,33,38,31,37,28,34,26,36,25,39,32,18);(1,2,3,4)(10,17)(12,15)(13,30)(14,29)(18,34)(19,21)(20,22)(23,35)(24,37)(25,33)(26,31)(27,32)(36,39)(38,40);(6,8)(9,11)(10,18)(12,25)(13,31)(14,32)(15,33)(16,28)(17,34)(19,36)(20,38)(21,39)(22,40)(23,37)(24,35)(26,30)(27,29);(9,12,16,15)(10,19,23,22)(11,25,28,33)(13,29,14,30)(17,20,35,21)(18,36,37,40)(24,39,34,38)(26,31,27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,d^2=i,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^b=di,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

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401701);