Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#1521 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
537 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 537 conjugacy classes of subgroups — too many to draw. Only its 191 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)(5,11,13,8)(6,9)(7,10)(12,17,14,16)(15,18)(19,20)(21,27)(22,30)(23,25)(24,29)(26,32)(28,34)(31,33);(3,5)(4,8)(6,14)(7,13)(9,17)(10,11)(12,15)(16,18)(19,21)(20,24)(22,31)(23,28)(25,33)(26,34)(27,30)(29,32);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(20,25)(21,28)(26,30)(29,31)64 conjugacy classes — view the full character table.
| maximal subgroup of | #107534, #107620, #107646, … |
| maximal normal subgroup of | #107534, #107620, #107646, … |
| maximal quotient of | C2×#7580, C2×#7581, C2×#7584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10554);