#6529024

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499844
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

2806 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)(19,27)(20,24)(21,34)(22,25)(23,30)(26,37)(28,46)(29,43)(31,45)(32,41)(33,44)(35,39)(36,42)(38,47)(40,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,31)(21,35)(22,29)(24,40)(25,39)(26,43)(27,34)(28,32)(30,37)(33,47)(36,48)(38,41)(42,46)(44,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20,22,32)(18,24,25,41)(19,28,29,31)(21,36,37,33)(23,38,39,40)(26,44,34,42)(27,45,43,46)(30,48,35,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(20,32)(24,41)(28,31)(33,36)(38,40)(42,44)(45,46)(47,48);(2,8)(6,9)(7,16)(14,15)(18,25)(19,29)(24,41)(26,34)(28,31)(30,35)(42,44)(47,48)

Polycyclic presentation:

2^9|c^2=h,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,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, 6436220);