#489438

order	512
nilpotency class	4
isoclinism	isoclinic class of #489390
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	C₄×C₈	C₂⁴
prime radical	#62433	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

698 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)(3,9,10,16)(4,12,13,7)(6,14,15,11)(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);(4,13)(7,12)(9,16)(11,14)(17,18)(19,25)(20,24)(21,37)(22,27)(23,29)(26,35)(28,38)(30,43)(31,39)(32,41)(33,45)(34,44)(36,40)(42,47)(46,48);(1,3)(2,6)(4,14)(5,10)(7,16)(8,15)(9,12)(11,13)(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);(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|a^2=h,d^2=g,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dgi,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396634);