Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #34967 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,10,13,8)(4,5,15,6)(11,14,16,12)(17,18,21,23)(19,25,27,22)(20,28,30,24)(26,31,29,32)(33,34,37,40)(35,41,43,38)(36,44,46,39)(42,47,45,48);(1,3)(2,8)(4,11)(5,12)(6,14)(7,13)(9,10)(15,16)(17,19)(18,22)(20,29)(21,27)(23,25)(24,32)(26,30)(28,31)(33,35)(34,38)(36,45)(37,43)(39,48)(40,41)(42,46)(44,47);(1,4)(2,5)(3,11)(6,9)(7,15)(8,12)(10,14)(13,16)(18,23)(20,30)(22,25)(26,29)(33,36,37,46)(34,39,40,44)(35,42,43,45)(38,47,41,48);(1,5,7,6)(2,4,9,15)(3,12,13,14)(8,11,10,16)(17,20,21,30)(18,24,23,28)(19,26,27,29)(22,31,25,32)(34,40)(35,43)(39,44)(42,45);(3,13)(8,10)(11,16)(12,14)(18,23)(19,27)(24,28)(26,29)(34,40)(38,41)(39,44)(47,48);(1,6,7,5)(2,4,9,15)(3,14,13,12)(8,11,10,16)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012111);