Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
| direct product | C8×#394 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | C23 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C23×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(13,14,15,17,16,18,19,20);(1,2)(3,4)(10,11);(5,6)(7,8)(9,10)(11,12);(2,4)(6,8)(9,11,12,10)176 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#2408, #72322182, #72322191 |
| maximal normal subgroup of | C8×#2408, #72322182, #72322191 |
| maximal quotient of | #72322660 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266268);