Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #482 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2033 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2)(3,4)(5,6,10,16,12,17,11,36)(7,18,22,50,24,51,23,60)(8,25,28,44,31,47,30,14)(9,32,34,46,35,65,19,15)(13,37,39,59,40,52,49,21)(20,53,55,68,57,41,56,48)(26,62,64,67,54,38,58,66)(27,43,33,45,29,42,63,61);(5,7,19,52,11,23,35,59,12,24,34,37,10,22,9,21)(6,13,32,50,16,39,46,51,17,40,65,60,36,49,15,18)(8,26,63,57,30,58,29,55,31,54,33,20,28,64,27,56)(14,41,61,62,25,48,43,67,44,53,45,38,47,68,42,66);(2,4)(5,8)(6,14)(7,20)(9,33)(10,30)(11,28)(12,31)(13,38)(15,45)(16,47)(17,44)(18,48)(19,29)(21,58)(22,56)(23,55)(24,57)(25,36)(26,52)(27,34)(32,43)(35,63)(37,54)(39,67)(40,62)(41,50)(42,65)(46,61)(49,66)(51,68)(53,60)(59,64)Polycyclic presentation:
2^9|a^2=g,b^2=dh,d^2=hi,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,d^c=dhi,e^c=ei,f^c=fh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55625);