#489326

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
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 ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

686 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)(3,12)(4,9)(5,10)(6,11)(7,14)(8,13)(15,16)(18,23)(20,30)(21,33)(24,38)(25,37)(26,41)(28,43)(29,42)(31,35)(32,46)(34,47)(36,39)(40,48)(44,45);(1,3)(2,7)(4,13)(5,14)(6,15)(8,11)(9,12)(10,16)(17,18)(19,24)(20,31)(21,35)(22,25)(23,30)(26,33)(27,39)(28,44)(29,45)(32,48)(34,41)(36,46)(37,47)(38,43)(40,42);(1,3)(2,8)(4,13)(5,14)(6,15)(7,11)(9,16)(10,12)(17,19,22,27)(18,24,25,39)(20,32,34,28)(21,29,33,42)(23,36,37,38)(26,40,35,45)(30,46,47,43)(31,48,41,44);(1,4,6,5)(2,9,11,10)(3,13,15,14)(7,12,8,16)(19,27)(24,39)(28,32)(29,42)(36,38)(40,45)(43,46)(44,48)

Polycyclic presentation:

2^9|c^2=i,d^2=hi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fhi,g^a=gi,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, 396522);