#490683

order	512
nilpotency class	4
isoclinism	isoclinic class of #490479
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19222	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

410 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)(18,24)(19,30)(20,34)(21,22)(25,35)(26,42)(27,38)(28,40)(29,39)(31,46)(32,36)(33,48)(37,44)(45,47);(17,18,25,39,22,29,20,24,23,27,34,40,21,28,35,38)(19,31,41,42,30,45,33,47,32,26,43,46,36,44,48,37);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,26)(20,33)(21,36)(22,30)(23,32)(24,37)(25,41)(27,31)(28,45)(29,44)(34,43)(35,48)(38,47)(39,46)(40,42);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,27)(19,32)(24,38)(28,29)(30,36)(33,48)(39,40)(41,43)

Polycyclic presentation:

2^9|b^2=egi,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,c^b=ci,d^b=di,d^c=di,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, 397879);