Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#1544 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
497 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 497 conjugacy classes of subgroups — too many to draw. Only its 207 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) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,6,10,7,9,15,17)(5,11,12,18,13,8,14,16)(19,20)(21,28)(22,26)(23,31)(24,27)(25,29)(30,33)(32,34);(3,5,7,13)(4,8,9,11)(6,14,15,12)(10,18,17,16)(19,21,24,30)(20,25,27,34)(22,29,23,32)(26,33,31,28);(19,22,24,23)(20,26,27,31)(21,29,30,32)(25,33,34,28)76 conjugacy classes — view the full character table.
| maximal subgroup of | #106039, #107353, #107360, … |
| maximal normal subgroup of | #106039, #107353, #107360, … |
| maximal quotient of | C2×#7564, C2×#7568, #160437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10687);