#6673601

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
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	#525	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1626 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)(19,24)(20,34)(21,26)(22,28)(23,29)(25,36)(27,40)(30,42)(31,45)(32,43)(33,44)(35,47)(37,46)(38,41)(39,48);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,33)(18,24,29,44)(20,35,30,38)(21,37,39,36)(22,31,40,32)(25,26,46,48)(27,43,28,45)(34,47,42,41);(17,20,23,30)(18,25,29,46)(19,21,33,39)(22,37,40,36)(24,41,44,47)(26,43,48,45)(27,42,28,34)(31,35,32,38);(2,7)(6,9)(8,15)(14,16);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21,23,39)(18,26,29,48)(19,30,33,20)(22,38,40,35)(24,42,44,34)(25,45,46,43)(27,47,28,41)(31,37,32,36)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,e^2=fi,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,e^c=ei,g^c=gi,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, 6580797);