Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#2115 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22⋊C4 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
577 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 577 conjugacy classes of subgroups — too many to draw. Only its 263 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);(3,4,6,11,7,12,18,26,8,10,19,23,20,24,31,33)(5,13,15,25,16,27,29,34,14,9,17,21,28,22,30,32);(3,5)(4,9)(6,17)(7,16)(8,14)(10,13)(11,25)(12,22)(15,19)(18,30)(20,28)(21,23)(24,27)(26,34)(29,31)(32,33);(5,14)(9,13)(15,17)(16,28)(21,25)(22,27)(29,30)(32,34)88 conjugacy classes — view the full character table.
| maximal subgroup of | #126957, #126958, #126959, … |
| maximal normal subgroup of | #126957, #126958, #126959, … |
| maximal quotient of | C2×#11419, #479985, C2×#11441, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26324);