Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#2098 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | #399 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
505 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 505 conjugacy classes of subgroups — too many to draw. Only its 167 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);(3,4,7,12,8,13,9,23)(5,14,16,26,18,30,17,10)(6,19,21,29,22,11,24,28)(15,31,20,34,32,27,33,25);(3,5,9,16,8,18,7,17)(4,10,23,14,13,26,12,30)(6,20,24,32,22,33,21,15)(11,27,29,25,19,31,28,34);(3,6)(4,11)(5,15)(7,21)(8,22)(9,24)(10,25)(12,28)(13,19)(14,31)(16,20)(17,33)(18,32)(23,29)(26,34)(27,30)52 conjugacy classes — view the full character table.
| maximal subgroup of | #137214, #137215, #137216, … |
| maximal normal subgroup of | #137214, #137215, #137216, … |
| maximal quotient of | C2×#10659, C2×#10660, C2×#10661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16727);