#501069

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

Polycyclic presentation:

2^9|a^2=fi,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, 408265);