Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C212 |
| direct product | C23×#342 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C22⋊C4 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C27 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C27 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
18818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4);(5,6);(11,12)(13,17,18,15)(14,16);(7,8,9,10)(11,13)(12,15)(14,18)(16,17)176 conjugacy classes — view the full character table.
| maximal subgroup of | #72234290, C22×#10965, C23×#2021, … |
| maximal normal subgroup of | #72234290, C22×#10965, C23×#2021, … |
| maximal quotient of | #72233872, #72235072, #72235720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276953);