#494503

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

1316 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)(9,10,12,17,28,34,32,31,15,20,13,18,14,19,29,27)(11,21,23,40,33,39,30,35,22,36,24,37,25,38,26,16);(1,2,3,4)(9,11)(10,16)(12,26)(13,30)(14,33)(15,22)(17,38)(18,39)(19,40)(20,35)(21,27)(23,29)(24,32)(25,28)(31,36)(34,37);(6,8)(9,10,15,20)(11,16,22,35)(12,27,13,31)(14,34,28,19)(17,32,18,29)(21,30,36,26)(23,38,24,39)(25,40,33,37);(11,22)(16,35)(21,36)(23,24)(25,33)(26,30)(37,40)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^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, 401699);