Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #482 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2101 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(5,6,11,19,13,20,12,38)(7,21,26,51,27,55,39,41)(8,28,32,53,33,46,37,47)(9,18,34,17,35,52,10,16)(14,25,42,24,44,63,43,23)(15,31,48,30,50,36,49,57)(22,56,60,40,62,65,61,67)(29,66,59,68,64,45,58,54);(1,2)(3,4)(5,7,11,26,13,27,12,39)(6,14,19,42,20,44,38,43)(8,29,37,58,33,64,32,59)(9,25,34,24,35,63,10,23)(15,45,49,68,50,66,48,54)(16,41,18,21,17,51,52,55)(22,57,61,36,62,30,60,31)(28,65,47,40,46,56,53,67);(2,4)(5,8,10,36,12,32,35,57,13,33,34,31,11,37,9,30)(6,15,18,53,19,49,17,28,20,50,52,47,38,48,16,46)(7,22,25,64,26,61,24,59,27,62,63,29,39,60,23,58)(14,40,41,66,43,65,55,68,44,67,51,45,42,56,21,54)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=dh,d^2=hi,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fhi,h^b=hi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56353);