#502911

order	512
nilpotency class	4
isoclinism	isoclinic class of #501256
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 ◁ C₂×#446 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

847 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)(11,22)(12,27)(13,19)(14,30)(15,31)(16,35)(17,24)(18,26)(20,28)(21,23)(25,39)(29,36)(32,37)(33,38)(34,40);(2,5)(6,7)(9,11)(10,17)(12,28)(13,24)(14,29)(15,32)(16,33)(18,35)(19,22)(20,39)(21,38)(23,26)(25,34)(27,40)(30,37)(31,36);(9,12,28,24,16,29,15,33,13,25,34,11,23,32,14,26)(10,18,30,37,21,22,40,39,19,38,31,36,35,17,20,27);(1,3,4,8)(2,6,5,7)(10,19)(17,22)(18,38)(20,40)(21,35)(27,39)(30,31)(36,37)

Polycyclic presentation:

2^9|c^2=fh,d^2=ei,f^2=hi,g^2=hi,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=gh,h^b=hi,g^c=gi

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, 410107);