#502919

order	512
nilpotency class	4
isoclinism	isoclinic class of #501248
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

835 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410115);