Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
730 conjugacy classes of subgroups; only the 282 normal subgroups are shown.
This group has 730 conjugacy classes of subgroups — too many to draw. Only its 282 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,27,23,26,22,28)(19,24,30,40,32,39,31,41)(20,33,35,43,36,44,37,25)(29,42,34,47,45,48,46,38);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,21,30,23,32,22,31)(18,24,27,40,26,39,28,41)(20,34,35,45,36,46,37,29)(25,42,33,47,43,48,44,38);(3,7)(6,8)(10,14)(13,16)(17,20,22,35,23,36,21,37)(18,25,28,33,26,43,27,44)(19,29,31,34,32,45,30,46)(24,38,41,42,39,47,40,48);(1,3,4,7)(2,6,5,8)(18,26)(24,39)(25,43)(27,28)(33,44)(38,47)(40,41)(42,48)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=h,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,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, 290129);