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)(5,6,11,17)(7,19,8,14)(9,18,12,15)(10,25,27,33)(13,23,30,21)(16,24,34,28)(20,31,22,32)(26,35,36,29);(5,7,11,8)(6,13,17,30)(9,20,12,22)(10,26,27,36)(14,25,19,33)(15,29,18,35)(16,32,34,31)(21,24,23,28);(1,3,4,2)(5,8,11,7)(6,14,17,19)(9,20,12,22)(10,23,27,21)(13,25,30,33)(15,32,18,31)(16,29,34,35)(24,26,28,36);(5,9,11,12)(6,15,17,18)(7,20,8,22)(10,24,27,28)(13,29,30,35)(14,31,19,32)(16,33,34,25)(21,26,23,36)Polycyclic presentation:
2^8|a^2=f,b^2=h,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 | #150148, #152286, #152447, … |
| maximal normal subgroup of | #150148, #152286, #152447, … |
| maximal quotient of | #483508, #483797, #483832, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26448);