Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7515 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#7515 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1402 |
| commutator | #127 | C22×C4 |
| Frattini, cosocle | #493 | C23 |
| prime radical | C2×#1619 | C2 |
| LCS | G ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×#493 ◁ G | |
| LECS | G ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
538 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 538 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(3,4)(5,12,13,8)(6,10,16,9)(7,11)(14,17,15,18);(4,8)(9,17)(10,12)(11,18)44 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#153613, C2×#153461, C2×#153617, … |
| maximal normal subgroup of | C2×#153613, C2×#153461, C2×#153617, … |
| maximal quotient of | #72279830, #72280001, #72280066 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60629);