#486679

order	512
nilpotency class	4
isoclinism	isoclinic class of #486320
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	C₂²×C₈	C₂⁴
prime radical	#62274	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=g,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dgi,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393875);