#489306

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
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

688 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)(18,24)(19,29)(20,32)(21,22)(25,41)(26,39)(27,33)(28,40)(30,37)(31,34)(35,36)(38,43)(42,45)(44,47);(1,3)(2,7)(4,13)(5,14)(6,15)(8,11)(9,12)(10,16)(17,18,23,26)(19,25,30,44)(20,33,36,40)(21,28,22,27)(24,35,39,32)(29,45,37,43)(31,47,34,41)(38,46,42,48);(1,3)(2,8)(4,13)(5,14)(6,15)(7,11)(9,16)(10,12)(17,19,23,30)(18,25,26,44)(20,34,36,31)(21,37,22,29)(24,38,39,42)(27,45,28,43)(32,48,35,46)(33,41,40,47);(1,4,6,5)(2,9,11,10)(3,13,15,14)(7,12,8,16)(18,26)(19,30)(24,39)(27,28)(29,37)(31,34)(33,40)(46,48)

Polycyclic presentation:

2^9|b^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,d^b=di,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, 396502);