Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11502 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#11502 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C24×D4 | C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
2346 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 2346 conjugacy classes of subgroups — too many to draw. Only its 234 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 — 4 generators on 26 points:
(1,2);(3,4,5,8,6,7,9,10)(11,12)(13,19)(14,22,23,17)(15,18)(16,21,24,26)(20,25);(11,13)(12,16)(14,21)(15,20)(17,19)(18,24)(22,25)(23,26);(4,7)(8,10)(13,20)(16,24)(19,25)(21,26)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72236808, #72236810, C2×#134045, … |
| maximal normal subgroup of | #72236808, #72236810, C2×#134045, … |
| maximal quotient of | C2×#149761, #72284262, C2×#149762 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387442);