Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#8285 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle, prime radical | C24 | C4×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(7,8,9,13,10,12,14,11)(15,16,18,21,19,17,20,22);(3,4,5,6)(8,11)(9,14)(12,13);(8,12)(11,13)(15,17,18,22,19,16,20,21)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#129171, C2×C8×#442, C2×#128691, … |
| maximal normal subgroup of | C2×#129171, C2×C8×#442, C2×#128691, … |
| maximal quotient of | C2×#96892, C2×#129171, C2×#129170 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90625);