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) = C24×C42 |
| direct product | C8×#418 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,7,9,8,10,11,12);(1,2,3,4)(13,14)(15,19)(16,18)(17,20);(14,18)(15,16)(19,20);(13,15,17,16)(14,19,20,18)176 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#3003, #72286522, #72320669, … |
| maximal normal subgroup of | C8×#3003, #72286522, #72320669, … |
| maximal quotient of | #72286304 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89598);