#486770

order	512
nilpotency class	4
isoclinism	isoclinic class of #486312
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

665 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)(19,29)(20,33)(21,22)(25,41)(26,34)(27,39)(28,40)(30,31)(32,37)(35,36)(38,43)(42,45)(44,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18,23,28)(19,25,32,44)(20,34,36,39)(21,27,22,26)(24,35,40,33)(29,45,37,43)(30,41,31,48)(38,47,42,46);(1,3,5,10)(2,6,8,12)(4,9,11,15)(7,13,14,16)(17,19,23,32)(18,25,28,44)(20,30,36,31)(21,37,22,29)(24,38,40,42)(26,45,27,43)(33,46,35,47)(34,41,39,48);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,20,21,35,23,36,22,33)(18,24,26,39,28,40,27,34)(19,30,37,47,32,31,29,46)(25,38,45,48,44,42,43,41)

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=dg,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393966);