Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C42 |
| direct product | C2×#17544 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C22×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
4696 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)(5,7)(6,8)(9,10)(12,16,17,18);(4,7)(5,10)(8,9)(11,12)(13,16)(14,18)(15,17);(4,8)(7,9);(11,13,15,14)(12,16,17,18)128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239587, #72239602, #72239674, … |
| maximal normal subgroup of | #72239587, #72239602, #72239674, … |
| maximal quotient of | #72239593, #72309237, C2×#185024 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250495);