Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#2129 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×D4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
615 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 615 conjugacy classes of subgroups — too many to draw. Only its 183 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 — 4 generators on 34 points:
(1,2);(3,4)(5,13,16,28)(6,10)(7,20,22,11)(8,12)(9,15)(14,29,17,31)(18,33,34,26)(19,27)(21,24)(23,30)(25,32);(3,5,8,16)(4,9,12,24)(6,17,19,14)(7,21,22,15)(10,25,27,23)(11,28,20,13)(18,30,34,32)(26,29,33,31);(3,6,8,19)(4,10,12,27)(5,14,16,17)(7,18,22,34)(9,23,24,25)(11,26,20,33)(13,29,28,31)(15,30,21,32)46 conjugacy classes — view the full character table.
| maximal subgroup of | #146657, #146672, #146699, … |
| maximal normal subgroup of | #146657, #146672, #146699, … |
| maximal quotient of | C2×#11434, C2×#11449, C2×#11482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26345);