Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | D8 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C4×D8 |
| prime radical | C23 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,16,11,17,9,15,12,18,26,34,25,33,8,14)(7,19,22,13,23,30,27,31,24,35,20,32,28,29,21,36);(1,2,3,4)(5,7)(6,13)(8,20)(9,22)(10,27)(11,28)(12,24)(14,29)(15,30)(16,35)(17,36)(18,32)(19,34)(21,26)(23,25)(31,33);(5,8,25,26,12,9,11,10)(6,14,33,34,18,15,17,16)(7,20,23,21,24,22,28,27)(13,29,31,19,32,30,36,35)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=deg,d^2=gh,e^2=g,g^2=h,b^a=bd,d^b=dg,e^b=eg,g^b=gh88 conjugacy classes — view the full character table.
| maximal subgroup of | #94052, #94053, #94054, … |
| maximal normal subgroup of | #94052, #94053, #94054, … |
| maximal quotient of | #129095, #129110, #129128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5658);