#6529464

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500226
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	#60703	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2668 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,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, 6436660);