Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(6,12)(7,16)(8,10)(9,17)(13,26)(14,29)(15,21)(18,31)(19,30)(20,32)(22,23)(24,27)(28,36)(33,35);(1,3,4,2)(5,6)(7,13)(8,21)(9,14)(10,12)(11,15)(16,27)(17,25)(18,33)(19,24)(20,28)(22,34)(23,29)(26,31)(30,36)(32,35);(5,7,17,31,10,19,23,32,11,20,22,30,8,18,9,16)(6,13,25,26,12,24,29,35,15,28,34,36,21,33,14,27)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=eg,d^2=h,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152438, #152439, #152443, … |
| maximal normal subgroup of | #152438, #152439, #152443, … |
| maximal quotient of | #140345, #140367, #140426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6159);