#489558

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	#62249	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

710 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)(3,11)(4,10)(5,8)(6,9)(7,13)(12,15)(14,16)(17,18,23,27)(19,28,31,47)(20,25,22,37)(21,33,36,45)(24,39,42,30)(26,32,44,35)(29,46,38,43)(34,40,48,41);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,16)(10,11)(17,19,23,31)(18,24,27,42)(20,29,22,38)(21,34,36,48)(25,40,37,41)(26,43,44,46)(28,45,47,33)(30,32,39,35);(2,8)(4,5)(7,15)(9,10)(11,16)(13,14)(17,20,23,22)(18,25,27,37)(19,29,31,38)(21,32,36,35)(24,40,42,41)(26,33,44,45)(28,46,47,43)(30,34,39,48);(2,9)(3,12)(8,10)(13,14)(17,21,20,32,23,36,22,35)(18,26,37,45,27,44,25,33)(19,30,38,48,31,39,29,34)(24,28,40,46,42,47,41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=ghi,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, 396754);