Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1862 |
| 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 | |
452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(5,6,11,17,34,50,35,51,12,18,9,15,10,16,36,52)(7,19,23,41,22,53,33,13,25,40,21,39,24,49,37,42)(8,26,28,60,29,48,55,46,32,47,30,61,31,14,43,45)(20,54,27,38,58,44,63,66,57,68,59,65,56,67,62,64);(1,2)(3,4)(5,7,10,22,12,25,34,24)(6,13,16,39,18,42,50,41)(8,27,29,62,32,59,31,63)(9,33,35,21,11,37,36,23)(14,44,26,54,48,67,47,68)(15,49,51,19,17,53,52,40)(20,55,56,30,57,43,58,28)(38,60,64,46,65,61,66,45);(2,4)(5,8,9,28,34,29,36,55,12,32,11,30,10,31,35,43)(6,14,15,45,50,26,52,60,18,48,17,46,16,47,51,61)(7,20,21,27,22,58,37,63,25,57,23,59,24,56,33,62)(13,38,19,44,39,66,53,68,42,65,40,67,41,64,49,54)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fhi,g^b=gh,h^b=hi,e^c=ei,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, 56240);