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×#12700 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
3550 conjugacy classes of subgroups; only the 354 normal subgroups are shown.
This group has 3550 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)(7,8,11,15,12,16,9,17)(10,14,18,21,20,22,19,13);(7,9,12,11)(8,13,16,21)(10,19,20,18)(14,15,22,17);(4,6)80 conjugacy classes — view the full character table.
| maximal subgroup of | #72266849, #72266854, C2×#499878, … |
| maximal normal subgroup of | #72266849, #72266854, C2×#499878, … |
| maximal quotient of | #72266860, #72266889, C2×#147565 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400572);