Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#1395 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C22 | #342 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
383 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 383 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) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(3,4)(5,13,15,25)(6,17,18,10)(7,20)(8,12)(9,14)(11,21)(16,30,29,26)(19,24,32,31)(22,34)(23,28)(27,33);(3,5,7,16,8,15,21,29)(4,9,11,22,12,23,20,33)(6,14,19,27,18,28,32,34)(10,13,24,26,17,25,31,30)52 conjugacy classes — view the full character table.
| maximal subgroup of | #146358, #146359, C4×#1395, … |
| maximal normal subgroup of | #146358, #146359, C4×#1395, … |
| maximal quotient of | C2×#7190, #126955, #126981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5576);