#6673586

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667748
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	#60578	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|b^2=fi,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,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, 6580782);