Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#1558 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
591 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 591 conjugacy classes of subgroups — too many to draw. Only its 207 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) = 128 |
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)(15,17)(16,20)(18,24)(21,26)(23,25);(4,7)(5,9)(11,13,15,21)(12,16,17,25)(14,23,24,20)(18,26,22,19);(12,17)(16,25)(18,22)(19,26)52 conjugacy classes — view the full character table.
| maximal subgroup of | #106067, #107302, #107340, … |
| maximal normal subgroup of | #106067, #107302, #107340, … |
| maximal quotient of | C2×#7595, C2×#7605, #160564, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10720);