#491146

order	512
nilpotency class	4
isoclinism	isoclinic class of #491091
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	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

408 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(18,24)(19,30)(20,22)(21,35)(25,36)(26,42)(27,39)(28,38)(29,40)(31,46)(32,48)(33,34)(37,44)(45,47);(17,18,25,38,22,28,21,24,23,29,35,39,20,27,36,40)(19,31,41,42,30,45,32,47,33,26,43,46,34,44,48,37);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,23,33)(18,26,29,31)(20,34,22,30)(21,32,36,48)(24,37,40,47)(25,41,35,43)(27,45,28,44)(38,46,39,42);(2,7)(6,15)(8,12)(9,16)(17,20,23,22)(18,27,29,28)(19,30,33,34)(21,25,36,35)(24,38,40,39)(26,44,31,45)(32,43,48,41)(37,42,47,46)

Polycyclic presentation:

2^9|b^2=egi,c^2=h,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,c^b=ci,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, 398342);