Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#3293 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1299 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1299 conjugacy classes of subgroups — too many to draw. Only its 487 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 — 5 generators on 34 points:
(1,2);(3,4,10,15)(5,16,19,21)(6,18,11,25)(7,12,24,30)(8,14,9,13)(17,32,34,28)(20,33,27,22)(23,31,26,29);(3,5,10,19)(4,11,15,6)(7,22,24,33)(8,25,9,18)(12,28,30,32)(13,21,14,16)(17,26,34,23)(20,29,27,31);(3,6,10,11)(4,5,15,19)(7,20,24,27)(8,21,9,16)(12,17,30,34)(13,18,14,25)(22,31,33,29)(23,32,26,28);(3,7)(4,12)(5,17)(6,20)(8,26)(9,23)(10,24)(11,27)(13,31)(14,29)(15,30)(16,32)(18,33)(19,34)(21,28)(22,25)58 conjugacy classes — view the full character table.
| maximal subgroup of | #131833, #131838, #131845, … |
| maximal normal subgroup of | #131833, #131838, #131845, … |
| maximal quotient of | C2×#19887, C2×#19894, C2×#19902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54092);