Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C8×#466 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#466 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,7,9,8,10,11,12);(1,2)(3,4)(14,17)(15,18)(19,20);(13,14,18,20,16,19,15,17);(2,4)176 conjugacy classes — view the full character table.
| maximal subgroup of | #72286666, #72286742, #72287158, … |
| maximal normal subgroup of | #72286666, #72286742, #72287158, … |
| maximal quotient of | #72286742, #72286922, #72320529 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 117913);