Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7307 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#7326 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
994 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 64 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(4,7,8,10)(11,12)(13,19,20,16)(14,18)(15,17)(21,25,26,23)(22,24);(3,4)(5,7,9,10)(6,8)(11,13,14,21,15,20,22,26)(12,16,18,23,17,19,24,25)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#128179 |
| maximal normal subgroup of | C2×#128179 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35359);