#6489604

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470028
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2741 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396800);