Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#1718 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #127 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
367 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 143 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) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,7,10)(5,11,12,18)(6,14,15,9)(8,16,17,13)(19,20)(21,27)(22,24)(23,31)(25,29)(26,30)(28,33)(32,34);(3,5,6,13,7,12,15,16)(4,8,9,11,10,17,14,18)(20,24)(25,32)(26,31)(27,33);(5,12)(8,17)(11,18)(13,16)(19,21,23,29,22,28,30,34)(20,25,26,27,24,32,31,33)52 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1718, #213121, #213123, … |
| maximal normal subgroup of | C4×#1718, #213121, #213123, … |
| maximal quotient of | C2×#8821, C2×#9826, C2×#9828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12598);