#489667

order	512
nilpotency class	4
isoclinism	isoclinic class of #489385
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
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

650 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)(18,23)(20,31)(21,33)(24,38)(25,37)(26,41)(27,42)(28,45)(29,44)(32,35)(34,48)(36,39)(40,43)(46,47);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,18)(19,24)(20,32)(21,35)(22,25)(23,31)(26,33)(27,43)(28,46)(29,47)(30,39)(34,41)(36,42)(37,48)(38,45)(40,44);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19,22,30)(18,24,25,39)(20,27,34,28)(21,29,33,44)(23,36,37,38)(26,40,35,47)(31,42,48,45)(32,43,41,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,33,48,22,34,21,31)(18,23,35,41,25,37,26,32)(19,27,44,45,30,28,29,42)(24,36,47,46,39,38,40,43)

Polycyclic presentation:

2^9|c^2=h,d^2=gh,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,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, 396863);