Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11505 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#12857 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2026 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 2026 conjugacy classes of subgroups — too many to draw. Only its 258 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)(13,16)(14,18,22,17)(15,19)(20,24,26,23)(21,25);(4,7)(8,10)(11,13,15,21)(12,16,19,25)(14,20,22,26)(17,23,18,24);(3,5,6,9)(4,8,7,10)(12,17)(14,22)(16,23)(18,19)(20,26)(24,25)80 conjugacy classes — view the full character table.
| maximal subgroup of | #72235766, #72235771, C2×#144642, … |
| maximal normal subgroup of | #72235766, #72235771, C2×#144642, … |
| maximal quotient of | C2×#135486 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401058);