Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#12089 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
522 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 522 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,7,5,6,8,10,9)(11,12,15,19,16,20,13,21)(14,23,24,26,25,17,22,18);(4,8)(5,9)(12,17)(13,15)(14,24)(18,21)(19,26)(20,23)(22,25);(3,5,7,8,6,9,10,4)(11,13,16,15)(14,24,25,22)(17,23)(18,26)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#483961, #72216740, #72216759, … |
| maximal normal subgroup of | C2×#483961, #72216740, #72216759, … |
| maximal quotient of | C2×#140188, C2×#140751, C22×#12089, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391031);