#6529054

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499849
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	#62879	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ 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) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6436250);