#6686106

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
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	#60707	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1637 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)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,23)(20,33)(21,26)(22,27)(24,36)(25,35)(28,45)(29,40)(30,31)(32,41)(34,39)(37,43)(38,44)(42,46)(47,48);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,22,29)(18,23,27,40)(20,34,38,47)(21,31,35,32)(24,42,43,45)(25,41,26,30)(28,36,46,37)(33,39,44,48);(17,20,35,37,22,38,21,36)(18,24,26,44,27,43,25,33)(19,28,32,47,29,46,31,34)(23,39,30,42,40,48,41,45);(2,7)(6,9)(8,15)(14,16)(19,29)(23,40)(28,46)(30,41)(31,32)(34,47)(39,48)(42,45);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,18)(19,30)(20,24)(21,25)(22,27)(23,31)(26,35)(28,42)(29,41)(32,40)(33,36)(34,39)(37,44)(38,43)(45,46)(47,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,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, 6593302);