Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#2180 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #442 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
323 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 151 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)(5,15,17,26)(6,19)(7,11)(8,23)(9,14)(10,16,27,33)(12,20)(13,25)(18,30,24,32)(21,31,34,28)(22,29);(3,5)(4,10)(6,16)(7,21)(8,24)(9,17)(11,26)(12,30)(13,31)(14,27)(15,29)(18,25)(19,32)(20,33)(22,34)(23,28);(3,6,8,7,9,20,25,22)(4,11,13,19,14,29,23,12)(5,16,24,21,17,33,18,34)(10,26,31,32,27,15,28,30)46 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#2180, #493252, #494995, … |
| maximal normal subgroup of | C4×#2180, #493252, #494995, … |
| maximal quotient of | C2×#12065, C2×#12066, #483591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26454);