Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#1664 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
447 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 447 conjugacy classes of subgroups — too many to draw. Only its 119 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);(3,4,8,12)(5,13,15,18)(6,11,7,10)(9,16,17,14)(19,20,24,22)(21,27,30,34)(23,31,33,26)(25,32,28,29);(3,5)(4,9)(6,16)(7,14)(8,15)(10,18)(11,13)(12,17)(19,21,24,30)(20,25,22,28)(23,32,33,29)(26,34,31,27);(3,5)(4,9)(6,14)(7,16)(8,15)(10,13)(11,18)(12,17)(19,22,24,20)(21,28,30,25)(23,26,33,31)(27,32,34,29)40 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1664, #209158, #209159, … |
| maximal normal subgroup of | C4×#1664, #209158, #209159, … |
| maximal quotient of | C2×#7788, C2×#7764, C2×#7765, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11974);