#6529583

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500157
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

2747 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:

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

Polycyclic presentation:

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

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, 6436779);