#502753

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×#436 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

857 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 409949);