Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19745 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C8 |
| direct product | C2×#19745 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2998 conjugacy classes of subgroups; only the 642 normal subgroups are shown.
This group has 2998 conjugacy classes of subgroups — too many to draw. Only its 642 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(4,7)(5,9)(8,10)(12,15)(13,17)(16,18);(3,4,5,7,6,8,9,10);(11,12,13,15,14,16,17,18);(4,8)(7,10)(12,16)(15,18)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19745⋊C3, #72214074, C22×#19745, … |
| maximal normal subgroup of | C2×#19745⋊C3, #72214074, C22×#19745, … |
| maximal quotient of | C2×#212918, C2×#213216, C2×#233521, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264960);