#489690

order	512
nilpotency class	4
isoclinism	isoclinic class of #489398
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	#62367	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

653 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)(17,18,24,29)(19,30,34,48)(20,26,23,38)(21,28,35,45)(22,27,36,46)(25,40,43,33)(31,42,39,47)(32,44,37,41);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19)(18,25)(20,31)(21,33)(22,37)(23,39)(24,34)(26,41)(27,30)(28,47)(29,43)(32,36)(35,40)(38,44)(42,45)(46,48);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,20,24,23)(18,26,29,38)(19,31,34,39)(21,36,35,22)(25,41,43,44)(27,28,46,45)(30,47,48,42)(32,40,37,33);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,36,24,35,20,22)(18,27,26,45,29,46,38,28)(19,32,31,33,34,37,39,40)(25,42,44,30,43,47,41,48)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=ghi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dgi,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396886);