Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,8,4,9)(3,11,12,16,13,7,14,15)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31);(1,3,6,13)(2,7,8,11)(4,12,5,14)(9,15,10,16)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(2,8)(7,11)(9,10)(15,16)(18,23)(22,31)(24,28)(25,32);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,15,11,16)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=gh,e^2=h,f^2=gh,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh64 conjugacy classes — view the full character table.
| maximal subgroup of | #106177, #107322, #107329, … |
| maximal normal subgroup of | #106177, #107322, #107329, … |
| maximal quotient of | #160455, #160474, #160477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10706);