#489563

order	512
nilpotency class	4
isoclinism	isoclinic class of #489373
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 ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

713 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)(3,11)(4,10)(5,8)(6,9)(7,13)(12,15)(14,16)(17,18)(19,31)(20,26)(21,38)(22,27)(23,42)(24,30)(25,33)(28,39)(29,40)(32,48)(34,43)(35,44)(36,45)(37,47)(41,46);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,16)(10,11)(17,19,24,20)(18,25,30,45)(21,36,39,33)(22,32,41,35)(23,37,40,34)(26,27,31,46)(28,47,38,43)(29,48,42,44);(2,8)(4,5)(7,15)(9,10)(11,16)(13,14)(17,20,24,19)(18,26,30,31)(21,32,39,35)(22,36,41,33)(23,37,40,34)(25,27,45,46)(28,44,38,48)(29,43,42,47);(2,9)(3,12)(8,10)(13,14)(17,21,23,22,24,39,40,41)(18,27,29,38,30,46,42,28)(19,32,34,36,20,35,37,33)(25,43,44,31,45,47,48,26)

Polycyclic presentation:

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

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, 396759);