#6529462

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

2755 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,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, 6436658);