Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8746 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#8748 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2498 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 2498 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)(5,9,10,7)(6,8)(11,12)(13,19,21,16)(14,22)(15,18)(17,23)(20,24,26,25);(11,13,15,21)(12,16,18,19)(14,20,23,26)(17,24,22,25);(3,5)(4,7)(6,10)(8,9)(11,14)(12,17,18,22)(13,20)(15,23)(16,24,19,25)(21,26)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#251828, C2×#133905, C2×#213860, … |
| maximal normal subgroup of | C2×#251828, C2×#133905, C2×#213860, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97294);