Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8346 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#8346 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3490 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,9,10,7)(6,8)(12,15,16,17);(3,5)(4,7)(6,10)(8,9)(11,12)(13,17)(14,16)(15,18);(3,4)(5,7)(6,8)(9,10)92 conjugacy classes — view the full character table.
| maximal subgroup of | #72219246, C2×#346446, #72232843, … |
| maximal normal subgroup of | #72219246, C2×#346446, #72232843, … |
| maximal quotient of | #72219244, #72219603, #72219616, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92439);