#489628

order	512
nilpotency class	4
isoclinism	isoclinic class of #489355
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	#466	D₄
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

658 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,23,28)(19,29,33,45)(20,26,35,46)(21,36,34,25)(22,39,38,27)(24,41,44,31)(30,43,37,47)(32,48,40,42);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19,23,33)(18,24,28,44)(20,31,35,41)(21,37,34,30)(22,40,38,32)(25,45,36,29)(26,42,46,48)(27,47,39,43);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,22,34,23,35,38,21)(18,25,27,26,28,36,39,46)(19,30,32,31,33,37,40,41)(24,42,43,45,44,48,47,29)

Polycyclic presentation:

2^9|a^2=i,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=dg,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396824);