#502883

order	512
nilpotency class	4
isoclinism	isoclinic class of #501076
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 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

859 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)(11,19)(12,25)(13,17)(14,29)(15,28)(16,21)(18,31)(20,22)(23,34)(24,35)(26,36)(27,39)(30,40)(32,37)(33,38);(2,5)(6,7)(9,11)(10,16)(12,26)(13,21)(14,32)(15,24)(17,19)(18,35)(20,29)(22,30)(23,31)(25,38)(27,33)(28,34)(36,39)(37,40);(9,12,25,10,15,29,30,18,13,27,39,17,31,40,14,28)(11,20,36,34,24,33,32,16,21,37,38,35,23,26,22,19);(1,3,4,8)(2,6,5,7)(10,17)(12,27)(16,19)(18,28)(20,37)(26,33)(29,40)(34,35)

Polycyclic presentation:

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

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, 410079);