Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#2179 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C4×D8 |
| prime radical | C2×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
327 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 159 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);(3,4,8,13,9,15,25,14)(5,16,18,31,20,33,19,27)(6,12,21,30,22,11,7,23)(10,26,28,34,29,24,32,17);(3,5)(4,10)(6,17)(7,24)(8,18)(9,20)(11,27)(12,31)(13,28)(14,32)(15,29)(16,23)(19,25)(21,26)(22,34)(30,33);(3,6,8,21,9,22,25,7)(4,11,13,23,15,12,14,30)(5,17,18,26,20,34,19,24)(10,27,28,16,29,31,32,33)64 conjugacy classes — view the full character table.
| maximal subgroup of | #136683, #136684, #140752, … |
| maximal normal subgroup of | #136683, #136684, #140752, … |
| maximal quotient of | C2×#12043, C2×#12044, C2×#12045, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26450);