#489278

order	512
nilpotency class	4
isoclinism	isoclinic class of #488895
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	#61485	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

701 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)(17,18,23,29)(19,30,31,45)(20,24,37,41)(21,28,22,27)(25,38,42,33)(26,36,40,34)(32,44,35,43)(39,47,46,48);(1,3)(2,7)(4,13)(5,14)(6,15)(8,11)(9,12)(10,16)(18,24)(20,34)(21,22)(25,39)(26,28)(27,40)(29,41)(30,44)(32,47)(33,38)(35,48)(36,37)(42,46)(43,45);(1,3)(2,8)(4,13)(5,14)(6,15)(7,11)(9,16)(10,12)(17,19,23,31)(18,25,29,42)(20,35,37,32)(21,38,22,33)(24,39,41,46)(26,44,40,43)(27,45,28,30)(34,48,36,47);(1,4,6,5)(2,9,11,10)(3,13,15,14)(7,12,8,16)(19,31)(25,42)(30,45)(32,35)(33,38)(39,46)(43,44)(47,48)

Polycyclic presentation:

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

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, 396474);