#486774

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	#2184	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

663 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:

(6,12)(13,16)(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);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18,23,29)(19,25,34,44)(20,24,36,40)(21,37,35,26)(22,27,38,28)(30,43,33,45)(31,39,46,48)(32,41,47,42);(1,3,5,10)(2,6,8,12)(4,9,11,15)(7,13,14,16)(17,19)(18,25)(20,31)(21,32)(22,33)(23,34)(24,39)(26,42)(27,45)(28,43)(29,44)(30,38)(35,47)(36,46)(37,41)(40,48);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(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|b^2=i,c^2=h,d^2=gh,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, 393970);