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 | #2092 |
| 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 | |
670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(2,4)(5,6)(7,18)(8,25,30,46)(9,32)(10,16)(11,36)(12,17)(13,21)(14,28,43,31)(15,19)(20,52,55,40)(22,49)(23,60)(24,50)(26,62,53,37)(27,42,29,41)(33,44,63,61)(34,45)(35,65)(38,58)(39,59)(47,54,68,56)(48,51)(57,66,64,67);(1,2,3,4)(5,7,19,51,11,23,35,59,12,24,34,58,10,22,9,21)(6,13,32,49,16,38,45,50,17,39,65,60,36,48,15,18)(8,26,63,55,31,57,29,54,30,53,33,20,28,64,27,56)(14,40,61,62,25,47,42,67,43,52,44,37,46,68,41,66);(5,8)(6,14)(7,20)(9,33)(10,31)(11,28)(12,30)(13,37)(15,44)(16,46)(17,43)(18,47)(19,29)(21,57)(22,56)(23,54)(24,55)(25,36)(26,51)(27,34)(32,42)(35,63)(38,67)(39,62)(40,49)(41,65)(45,61)(48,66)(50,68)(52,60)(53,58)(59,64)Polycyclic presentation:
2^9|a^2=g,b^2=dh,d^2=hi,e^2=i,f^2=hi,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, 55624);