Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11505 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12699 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C24×D4 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
4086 conjugacy classes of subgroups; only the 354 normal subgroups are shown.
This group has 4086 conjugacy classes of subgroups — too many to draw. Only its 354 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(7,8,10,14,11,15,19,22)(9,13,17,21,16,12,18,20);(8,12)(9,16)(13,15)(14,21)(17,18)(20,22);(4,6)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#493385, #72238534, #72238537, … |
| maximal normal subgroup of | C2×#493385, #72238534, #72238537, … |
| maximal quotient of | #72261792, #72261801, #72266738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400571);