#489274

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
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

717 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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,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, 396470);