#6524888

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	#62830	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2898 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)(17,18,22,28,23,26,21,27)(19,29,31,41,32,24,36,39)(20,33,34,40,35,42,37,25)(30,44,47,38,46,48,45,43);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,38)(26,29)(27,39)(28,41)(33,44)(34,45)(35,46)(37,47)(40,43)(42,48);(2,7)(3,9)(4,11)(12,16)(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,40)(27,33)(28,42)(29,43)(31,45)(32,46)(36,47)(39,44)(41,48);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,12,13,16)(18,26)(24,29)(25,40)(27,28)(33,42)(38,43)(39,41)(44,48);(3,9)(6,13)(10,15)(12,16)(17,21,23,22)(18,27,26,28)(19,31,32,36)(20,34,35,37)(24,39,29,41)(25,33,40,42)(30,45,46,47)(38,44,43,48)

Polycyclic presentation:

2^9|a^2=gh,d^2=h,e^2=i,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, 6432084);