Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(10,14)(11,18)(12,19)(15,21)(16,20)(17,24)(22,23);(2,4)(5,7,8,6)(10,15)(14,21)(17,22)(23,24);(9,10,16,24,12,17,11,14,13,15,18,23,19,22,20,21)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=egh,e^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #93938, #152453, #152804, … |
| maximal normal subgroup of | #93938, #152453, #152804, … |
| maximal quotient of | #140344, #140366, #140429, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6156);