Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349298 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62827 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,23,26)(19,29,31,40)(20,34,35,41)(21,28,22,27)(24,33,39,32)(25,36,42,37)(30,44,45,38)(43,46,48,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,24)(20,30)(21,33)(22,32)(23,31)(25,38)(26,39)(27,40)(28,29)(34,48)(35,45)(36,47)(37,46)(41,43)(42,44);(1,2)(3,7)(4,13)(5,8)(6,9)(10,11)(12,16)(14,15)(17,20)(18,25)(19,30)(21,37)(22,36)(23,35)(24,38)(26,42)(27,34)(28,41)(29,43)(31,45)(32,47)(33,46)(39,44)(40,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,26)(20,35)(24,39)(27,28)(29,40)(30,45)(36,37)(46,47);(3,12)(7,16)(10,15)(11,14)(19,31)(24,39)(29,40)(30,45)(32,33)(38,44)(43,48)(46,47)Polycyclic presentation:
2^9|a^2=i,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256529);