#508381

order	512
nilpotency class	4
isoclinism	isoclinic class of #508286
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13331	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

703 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)(3,6)(4,5)(7,8)(9,10,14,19,13,18,29,15)(11,20,23,31,22,16,24,33)(12,25,28,34,27,35,30,17)(21,37,39,32,38,26,40,36);(2,5)(6,8)(9,11)(10,16)(12,26)(13,22)(14,23)(15,31)(17,21)(18,20)(19,33)(24,29)(25,39)(27,37)(28,36)(30,32)(34,38)(35,40);(3,7)(6,8)(9,12)(10,17)(11,21)(13,27)(14,30)(15,25)(16,32)(18,34)(19,35)(20,36)(22,38)(23,40)(24,39)(26,31)(28,29)(33,37);(1,3,4,7)(2,6,5,8)(12,27)(17,34)(21,38)(25,35)(26,37)(28,30)(32,36)(39,40)

Polycyclic presentation:

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

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, 415577);