Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#2327 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1479 conjugacy classes of subgroups; only the 823 normal subgroups are shown.
This group has 1479 conjugacy classes of subgroups — too many to draw. Only its 823 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) = 256 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(4,6)(7,8,10,9)(12,16)(13,19)(14,22)(24,26);(7,9,10,8)(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25);(3,4,5,6)(11,13)(12,17)(14,21)(15,19)(16,20)(18,24)(22,25)(23,26);(11,14,15,22)(12,18,16,23)(13,21,19,25)(17,24,20,26)88 conjugacy classes — view the full character table.
| maximal subgroup of | #97027, #106067, #108065, … |
| maximal normal subgroup of | #97027, #106067, #108065, … |
| maximal quotient of | C2×#13779, C2×#13786, C2×#13790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27056);