Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8608 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8609 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1510 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);(4,7)(5,8)(9,10)(11,12)(13,17,18,15)(14,16);(3,4,8,10,6,9,5,7);(11,13)(12,15)(14,18)(16,17)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D8×#99, #72313705, #72313710, … |
| maximal normal subgroup of | C2×D8×#99, #72313705, #72313710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96001);