#489220

order	512
nilpotency class	4
isoclinism	isoclinic class of #488889
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	#61393	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

725 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396416);