#6529917

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500489
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1611 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:

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

Polycyclic presentation:

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

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, 6437113);