#486687

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	#62279	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

695 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)(19,28)(20,32)(21,22)(25,40)(26,45)(27,39)(29,30)(31,36)(33,37)(34,35)(38,42)(41,43)(44,48);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,18,23,27)(19,25,31,44)(20,33,35,45)(21,26,22,37)(24,34,39,32)(28,42,36,43)(29,40,30,48)(38,46,41,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,19)(18,25)(20,30)(21,36)(22,28)(23,31)(24,38)(26,43)(27,44)(29,35)(32,47)(33,48)(34,46)(37,42)(39,41)(40,45);(17,20,21,34,23,35,22,32)(18,24,37,45,27,39,26,33)(19,29,36,47,31,30,28,46)(25,41,42,48,44,38,43,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 393883);