#6524885

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
prime radical	#62835	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2864 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,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, 6432081);