#486670

order	512
nilpotency class	4
isoclinism	isoclinic class of #486306
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	#62258	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#435 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

732 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)(17,18,22,26)(19,27,29,46)(20,31,33,45)(21,36,34,25)(23,38,39,28)(24,41,43,35)(30,47,37,42)(32,44,48,40);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19,22,29)(18,23,26,39)(20,32,33,48)(21,37,34,30)(24,42,43,47)(25,44,36,40)(27,45,46,31)(28,41,38,35);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(20,33)(24,43)(27,46)(28,38)(31,45)(32,48)(35,41)(42,47);(17,20,34,41,22,33,21,35)(18,24,36,45,26,43,25,31)(19,28,37,48,29,38,30,32)(23,27,44,47,39,46,40,42)

Polycyclic presentation:

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

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, 393866);