Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#17546 |
| 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 | C2×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(4,8,9,10)(12,16)(13,14)(17,18);(3,4)(5,8)(6,10)(7,9)(11,12)(13,16)(14,18)(15,17);(11,13,15,14)(12,16,17,18);(3,5,7,6)(4,8,9,10)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#116398, C2×#117930, #72215258, … |
| maximal normal subgroup of | C2×#116398, C2×#117930, #72215258, … |
| maximal quotient of | C2×#208390, C2×#208392, #72239122 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250497);