Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12719 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C22 | #1782 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2544 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)(5,12,14,8)(6,10)(7,11)(9,13,16,18)(15,17);(4,8)(9,10)(11,12)(16,17);(3,5)(4,9,11,16)(6,13)(7,14)(8,10,12,17)(15,18)80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232831, C2×#11714269, C2×#494016, … |
| maximal normal subgroup of | #72232831, C2×#494016, #72257229, … |
| maximal quotient of | #72232818, C2×#146667, C2×#133995, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400638);