Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C213 |
| direct product | C22×#1446 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C26 | C23 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
21970 conjugacy classes of subgroups; only the 4418 normal subgroups are shown.
This group has 21970 conjugacy classes of subgroups — too many to draw. Only its 4418 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,4);(1,2)(3,4);(6,8)(9,10,11,12)(14,16);(13,14)(15,16)(17,18,20,22)(19,23,24,21);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,19,20,24)(18,21,22,23)176 conjugacy classes — view the full character table.
| maximal subgroup of | #72238817, #72238845, C22×#17309, … |
| maximal normal subgroup of | #72238817, #72238845, C22×#17309, … |
| maximal quotient of | C2×#358779, C2×#358792, C22×#7557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 421211);