Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#15258 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
21846 conjugacy classes of subgroups; only the 1322 normal subgroups are shown.
This group has 21846 conjugacy classes of subgroups — too many to draw. Only its 1322 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);(4,6)(8,10)(12,14)(16,18);(7,8,9,10)(15,16,17,18);(11,12)(13,14)(15,16)(17,18);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)116 conjugacy classes — view the full character table.
| Fitting subgroup of | #72238101 |
| maximal subgroup of | #72213999, D3×#15258, #72215608, … |
| maximal normal subgroup of | #72213999, #72215608, C2×#647656, … |
| maximal quotient of | #72214731, D3×#15258, C2×#417744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 515979);