#512106

order	512
nilpotency class	4
isoclinism	isoclinic class of #511755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33520	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

970 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)(10,16)(11,20)(12,24)(14,27)(15,19)(17,23)(18,26)(21,33)(22,30)(25,29)(28,31)(32,34)(35,38)(36,39);(2,5)(6,8)(9,10,13,16)(11,17,22,31)(12,18,25,32)(14,15,19,27)(20,28,30,23)(21,33,36,39)(24,34,29,26)(35,40,38,37);(1,3,4,7)(2,6,5,8)(9,11,13,22)(10,17,16,31)(12,21,25,36)(14,28,19,23)(15,30,27,20)(18,33,32,39)(24,37,29,40)(26,38,34,35);(9,12)(10,18)(11,21)(13,25)(14,29)(15,26)(16,32)(17,33)(19,24)(20,35)(22,36)(23,37)(27,34)(28,40)(30,38)(31,39)

Polycyclic presentation:

2^9|b^2=i,c^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,h^a=hi,f^c=fh,g^c=ghi,h^c=hi,f^d=fhi,g^d=ghi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419302);