#6674454

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1756 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)(3,11)(4,8)(5,10)(6,9)(7,13)(12,16)(14,15)(17,18)(19,31)(20,35)(21,30)(22,28)(23,29)(24,27)(25,33)(26,37)(32,43)(34,45)(36,42)(38,44)(39,47)(40,48)(41,46);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,23,33)(18,25,29,31)(20,36,38,40)(21,32,24,34)(22,39,41,37)(26,46,47,28)(27,43,30,45)(35,48,44,42);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20,23,38)(18,26,29,47)(19,22,33,41)(21,37,24,39)(25,42,31,48)(27,35,30,44)(28,43,46,45)(32,40,34,36);(2,9)(7,11)(8,10)(15,16)(17,21,23,24)(18,27,29,30)(19,32,33,34)(20,37,38,39)(22,40,41,36)(25,43,31,45)(26,35,47,44)(28,42,46,48);(1,5,6,4)(2,10,9,8)(3,12,13,14)(7,15,11,16)(17,22,23,41)(18,28,29,46)(19,20,33,38)(21,36,24,40)(25,44,31,35)(26,43,47,45)(27,48,30,42)(32,39,34,37)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^e=hi

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, 6581650);