Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7307 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
| direct product | C2×#7308 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
1562 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 1562 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 64 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,14,18,15,19,23,26)(13,17,20,25,21,16,22,24);(3,5)(4,7,8,9)(6,10)(12,16,19,17)(18,25,26,24)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72210062, #72210063, #72210064, … |
| maximal normal subgroup of | #72210062, #72210063, #72210064, … |
| maximal quotient of | #72262170, C22×#7308, #72266600, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35207);