Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 159 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,13)(8,16)(9,23)(10,14)(11,17)(12,19)(15,25)(18,32)(20,31)(21,29)(22,34)(24,30)(26,33)(27,36)(28,35);(5,7)(6,12)(8,18)(9,24)(10,21)(11,20)(13,23)(14,27)(15,29)(16,30)(17,28)(19,25)(22,35)(26,36)(31,33)(32,34);(1,3,4,2)(5,7)(6,13)(8,21)(9,19)(10,18)(11,20)(12,23)(14,32)(15,30)(16,29)(17,31)(22,36)(24,25)(26,35)(27,34)(28,33);(5,8,11,10)(6,14,17,16)(7,18,20,21)(9,22,26,25)(12,27,28,30)(13,29,31,32)(15,33,34,23)(19,24,35,36)Polycyclic presentation:
2^8|a^2=f,c^2=f,d^2=h,e^2=gh,g^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gh,e^b=eg,g^b=gh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #152443, #152444, #152531, … |
| maximal normal subgroup of | #152443, #152444, #152531, … |
| maximal quotient of | #483507, #483796, #483831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26447);