#6684549

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682628
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
prime radical	#61217	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1648 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18)(19,24)(20,33)(21,26)(22,28)(23,29)(25,32)(27,38)(30,39)(31,40)(34,42)(35,45)(36,43)(37,44)(41,46)(47,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,24)(20,22)(21,34)(23,32)(25,29)(26,42)(27,40)(28,33)(30,36)(31,38)(35,46)(37,47)(39,43)(41,45)(44,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20,22,32,23,31,38,19)(18,25,27,33,29,24,28,40)(21,34,35,46,37,47,36,30)(26,41,43,42,44,39,45,48);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16);(3,13)(7,10)(11,12)(15,16)(17,21,23,37)(18,26,29,44)(19,30,32,46)(20,34,31,47)(22,35,38,36)(24,39,25,41)(27,43,28,45)(33,42,40,48)

Polycyclic presentation:

2^9|c^2=fg,d^2=f,e^2=i,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,h^c=hi,h^e=hi

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, 6591745);