#501365

order	512
nilpotency class	4
isoclinism	isoclinic class of #501014
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33520	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

715 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)(3,6)(4,5)(7,8)(9,10,15,19,16,20,13,14)(11,17,31,35,23,33,22,29)(12,24,26,18,28,30,27,34)(21,36,37,32,39,25,38,40);(2,5)(6,8)(9,11)(10,17)(12,25)(13,22)(14,29)(15,31)(16,23)(18,21)(19,35)(20,33)(24,38)(26,40)(27,32)(28,36)(30,37)(34,39);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,21)(13,26)(14,30)(15,27)(16,28)(17,32)(19,24)(20,34)(22,37)(23,39)(25,29)(31,38)(33,40)(35,36);(2,5)(6,8)(9,13,16,15)(10,14,20,19)(11,22,23,31)(12,26,28,27)(17,29,33,35)(18,30,34,24)(21,37,39,38)(25,40,36,32)

Polycyclic presentation:

2^9|a^2=fi,c^2=e,d^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,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, 408561);