Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11078 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,6)(7,8,9,10)(13,17)(15,18);(12,15,16,18);(4,6)(11,12)(13,18)(14,16)(15,17)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#475764, C2×#11711335, #72215281, … |
| maximal normal subgroup of | C2×#475764, #72215281, #72215680, … |
| maximal quotient of | #72213342, #72215245, C2×#116398 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264936);