#6529316

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500002
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2757 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)(19,23)(20,30,32,44)(21,26)(22,25)(24,33,41,35)(27,45,31,48)(28,38)(29,39)(34,43)(36,42)(37,46,40,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,31)(21,34)(22,28)(24,40)(25,38)(26,43)(27,32)(29,36)(30,45)(33,46)(35,47)(37,41)(39,42)(44,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20,22,32)(18,24,25,41)(19,27,28,31)(21,35,36,33)(23,37,38,40)(26,44,42,30)(29,47,34,46)(39,48,43,45);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,25)(20,32)(23,38)(26,42)(27,31)(33,35)(39,43)(46,47);(2,8)(6,9)(7,16)(14,15)(18,25)(19,28)(24,41)(26,42)(27,31)(29,34)(30,44)(46,47)

Polycyclic presentation:

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

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, 6436512);