Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#1657 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×#496 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
505 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 505 conjugacy classes of subgroups — too many to draw. Only its 119 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);(4,7)(5,9)(8,10)(12,17)(13,20)(14,15)(19,22)(21,23)(24,25);(5,9)(7,10)(11,12)(13,18)(14,22)(15,17)(16,19)(20,25)(21,26)(23,24);(3,4)(5,10)(6,8)(7,9)(11,13,16,21)(12,18,19,26)(14,20,15,23)(17,24,22,25)40 conjugacy classes — view the full character table.
| maximal subgroup of | #152577, #153160, #153164, … |
| maximal normal subgroup of | #152577, #153160, #153164, … |
| maximal quotient of | C2×#7736, C2×#7738, C2×#7740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11970);