#501067

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3061	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1103 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)(9,10,14,20,13,19,28,35,15,21,30,38,16,22,31,32)(11,17,24,34,26,37,29,18,12,25,27,39,23,33,40,36);(2,4)(5,6)(7,8)(9,11)(10,17)(12,15)(13,26)(14,24)(16,23)(18,35)(19,37)(20,34)(21,25)(22,33)(27,30)(28,29)(31,40)(32,36)(38,39);(6,8)(9,12,15,11)(10,18,21,36)(13,26,16,23)(14,29,30,40)(17,32,25,35)(19,34,22,39)(20,37,38,33)(24,31,27,28);(2,4)(9,13,15,16)(10,19,21,22)(11,23,12,26)(14,28,30,31)(17,33,25,37)(18,34,36,39)(20,35,38,32)(24,40,27,29)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408263);