Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | #3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,11,6)(4,8)(5,7)(10,15,16,13)(12,14)(17,18)(19,26,28,31)(20,29)(21,24)(22,25)(23,27,32,30);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(2,7)(6,9)(8,14)(13,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26);(1,4,5,12)(2,8,7,14)(3,10,11,16)(6,13,9,15)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=g,e^2=h,b^a=be,c^a=cgh,e^a=eg,e^b=eh,f^b=fgh64 conjugacy classes — view the full character table.
| maximal subgroup of | #105902, #106210, #107440, … |
| maximal normal subgroup of | #105902, #106210, #107440, … |
| maximal quotient of | #160456, #160473, #160474, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10559);