#489675

order	512
nilpotency class	4
isoclinism	isoclinic class of #489394
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 ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

663 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,30)(20,26)(21,28)(22,37)(23,39)(24,29)(25,33)(27,36)(31,47)(32,44)(34,45)(35,46)(38,41)(40,42)(43,48);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19)(18,25)(20,31)(21,33)(22,38)(23,40)(24,34)(26,41)(27,45)(28,42)(29,43)(30,37)(32,36)(35,48)(39,44)(46,47);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,20,24,23)(18,26,29,39)(19,31,34,40)(21,36,35,22)(25,41,43,44)(27,46,37,28)(30,42,45,47)(32,48,38,33);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,36,24,35,20,22)(18,27,26,28,29,37,39,46)(19,32,31,33,34,38,40,48)(25,42,44,45,43,47,41,30)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396871);