Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2020 conjugacy classes of subgroups; only the 466 normal subgroups are shown.
This group has 2020 conjugacy classes of subgroups — too many to draw. Only its 466 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(6,8)(10,12)(13,14,17,19,18,22,16,21)(15,20,25,26,23,27,24,28);(1,2,3,4)(14,19)(16,17)(20,26)(21,22)(24,25)(27,28);(5,6,7,8)(13,15)(14,20)(16,24)(17,25)(18,23)(19,26)(21,28)(22,27);(9,10,11,12)(15,23)(20,27)(24,25)(26,28)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^b=ei,h^b=hi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301016);