#486686

order	512
nilpotency class	4
isoclinism	isoclinic class of #486327
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	#62278	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

727 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)(17,18)(19,30)(20,24)(21,28)(22,27)(23,29)(25,32)(26,34)(31,44)(33,45)(35,42)(36,40)(37,41)(38,43)(39,48)(46,47);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(18,24)(20,34)(21,22)(25,39)(26,28)(27,40)(29,41)(30,44)(31,47)(32,38)(35,48)(36,37)(42,45)(43,46);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,19)(18,25)(20,35)(21,38)(22,32)(23,33)(24,39)(26,44)(27,45)(28,30)(29,43)(31,37)(34,48)(36,47)(40,42)(41,46);(17,20,21,36,23,37,22,34)(18,26,27,41,29,40,28,24)(19,31,38,48,33,35,32,47)(25,42,45,39,43,44,30,46)

Polycyclic presentation:

2^9|a^2=hi,d^2=gi,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,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393882);