Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C23×C42 |
| direct product | C2×#7014 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C42 |
| commutator | C4 | C2×C82 |
| Frattini, cosocle | C43 | C23 |
| socle, prime radical | C24 | #95 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
806 conjugacy classes of subgroups; only the 386 normal subgroups are shown.
This group has 806 conjugacy classes of subgroups — too many to draw. Only its 386 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(3,4,5,7,6,8,9,10)(19,20)(21,27,29,33)(22,25)(23,26)(24,28,31,34)(30,32);(11,12,13,15,14,16,17,18)(19,21)(20,24,26,31)(22,28)(23,29)(25,27,32,33)(30,34)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7014⋊C3, #72210220, C2×#96658, … |
| maximal normal subgroup of | C2×#7014⋊C3, #72210220, C2×#96658, … |
| maximal quotient of | C2×#93249, C2×#93250, C2×#93254, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2058);