Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13673 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#13673 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | #127 | C24 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1298 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 1298 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 4 generators on 18 points:
(1,2);(6,14)(8,15)(16,17);(3,4)(5,8)(6,9)(7,10)(11,15)(12,16)(13,17)(14,18);(3,5)(4,8)(6,11)(7,13)(9,15)(10,17)(12,14)(16,18)50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#512862, C2×#153613, C2×#153617, … |
| maximal normal subgroup of | C2×#512862, C2×#153613, C2×#153617, … |
| maximal quotient of | C2×#153613, C2×#153617, #72279466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420049);