Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C27×C4 |
| direct product | C8×#386 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | C23 |
| commutator | C22 | C22×C4×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C23×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2578 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);(2,4)(5,6)(7,8)(9,10,11,12);(1,2,3,4)(10,12);(6,8)224 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#2847, #72320318, #72320342, … |
| maximal normal subgroup of | C8×#2847, #72320318, #72320342, … |
| maximal quotient of | #72286793, #72287665, #72321323, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261485);