#502907

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
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

840 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)(12,16)(13,19)(14,29)(15,31)(17,25)(18,23)(20,28)(22,38)(24,37)(26,35)(27,33)(30,36)(32,39)(34,40);(2,5)(6,7)(10,16)(11,22)(13,23)(14,27)(15,28)(17,34)(18,33)(19,29)(21,36)(24,30)(25,26)(32,35)(37,38)(39,40);(9,11,23,22,15,26,27,39,12,24,14,30,28,40,13,25)(10,17,19,34,20,36,29,37,16,32,33,35,31,38,18,21);(1,3,4,8)(2,6,5,7)(10,16)(17,32)(18,29)(19,33)(20,31)(21,37)(34,35)(36,38)

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=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410103);