#489688

order	512
nilpotency class	4
isoclinism	isoclinic class of #489398
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	C₄×C₈	C₂⁴
prime radical	#62368	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

677 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,23,28)(19,29,32,42)(20,25,22,38)(21,34,37,45)(24,40,43,31)(26,36,27,33)(30,46,39,47)(35,41,48,44);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19)(18,24)(20,30)(21,35)(22,39)(23,32)(25,41)(26,46)(27,47)(28,43)(29,34)(31,36)(33,40)(37,48)(38,44)(42,45);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,20,23,22)(18,25,28,38)(19,30,32,39)(21,36,37,33)(24,41,43,44)(26,34,27,45)(29,47,42,46)(31,48,40,35);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,20,33,23,37,22,36)(18,26,38,34,28,27,25,45)(19,31,39,48,32,40,30,35)(24,42,41,47,43,29,44,46)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=gh,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dgi,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fhi

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, 396884);