#6529916

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

Polycyclic presentation:

2^9|b^2=f,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, 6437112);