Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #380 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
441 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 441 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,15)(8,19,21,12)(9,22)(10,14)(11,17)(13,23)(16,29,31,27)(18,28)(20,26,35,34)(24,30,33,36)(25,32);(1,2)(3,4)(5,7,9,17,10,18,23,32)(6,11,13,15,14,25,22,28)(8,16,20,30,21,31,35,36)(12,24,26,27,19,33,34,29);(2,4)(5,8)(6,12)(7,16)(9,20)(10,21)(11,24)(13,26)(14,19)(15,27)(17,30)(18,31)(22,34)(23,35)(25,33)(28,29)(32,36)Polycyclic presentation:
2^8|a^2=g,b^2=d,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131332, #131333, #144718, … |
| maximal normal subgroup of | #131332, #131333, #144718, … |
| maximal quotient of | #116427, #116439, #116440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4086);