#490312

order	512
nilpotency class	4
isoclinism	isoclinic class of #490075
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	#33778	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

679 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397508);