Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8101 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
| direct product | C2×#8107 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#95 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,5,8,6,7,9,10)(11,12)(13,17)(14,16)(15,18);(4,7)(5,9)(11,13)(12,15,16,17)(14,18);(13,18)(15,17)104 conjugacy classes — view the full character table.
| maximal subgroup of | #72231659, #72231663, C2×#234827, … |
| maximal normal subgroup of | #72231659, #72231663, C2×#234827, … |
| maximal quotient of | #72309185 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88549);