#6529632

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500006
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62879	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2707 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,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, 6436828);