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) = C26 |
| direct product | C2×#2135 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×Q8 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
599 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 599 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);(3,4,8,13,9,14,6,11)(5,15,18,33,19,30,16,32)(7,21,20,31,23,12,24,29)(10,25,26,22,27,34,28,17);(3,5,9,19)(4,10,14,27)(6,18,8,16)(7,22,23,17)(11,28,13,26)(12,30,21,15)(20,25,24,34)(29,32,31,33);(3,6,9,8)(4,11,14,13)(5,16,19,18)(7,20,23,24)(10,26,27,28)(12,29,21,31)(15,32,30,33)(17,25,22,34)46 conjugacy classes — view the full character table.
| maximal subgroup of | #146652, #146672, #146699, … |
| maximal normal subgroup of | #146652, #146672, #146699, … |
| maximal quotient of | C2×#11434, C2×#11450, C2×#11481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26356);