Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#10683 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1364 conjugacy classes of subgroups; only the 394 normal subgroups are shown.
This group has 1364 conjugacy classes of subgroups — too many to draw. Only its 394 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12,15,16,14,18,13,17);(3,5,6,10)(4,7,8,9)(12,16)(13,15)(17,18)(19,20,22,25)(21,23,26,24);(12,16)(13,15)(17,18)(20,23)(21,26)(24,25)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10683⋊C3, C2×#244402, #72214409, … |
| maximal normal subgroup of | C2×#10683⋊C3, C2×#244402, #72214409, … |
| maximal quotient of | C2×#121387, C2×#117091, #72224918, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151145);