Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#1605 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
600 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 600 conjugacy classes of subgroups — too many to draw. Only its 135 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);(7,10)(11,12,15,18)(13,19,21,26)(14,22,24,17)(16,23,25,20);(3,4)(5,7)(6,8)(9,10)(11,13,15,21)(12,16,18,25)(14,23,24,20)(17,26,22,19);(3,5,6,9)(4,7,8,10)46 conjugacy classes — view the full character table.
| maximal subgroup of | #137325, #137327, #137346, … |
| maximal normal subgroup of | #137325, #137327, #137346, … |
| maximal quotient of | C2×#7763, C2×#7757, C2×#7772, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11099);