#489519

order	512
nilpotency class	4
isoclinism	isoclinic class of #489373
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	#62488	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

654 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=gi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gi,d^b=dg,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, 396715);