Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #20338 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C4 | C22×C42 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C22×#97 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,9,5,11,16,24,6,10,15,25,17,7,18,20)(3,8,13,22,14,23,28,32,12,21,26,31,27,19,29,30);(2,7,10,11)(4,15)(8,19,21,23)(9,24,25,20)(13,26)(16,18)(22,32,31,30)(28,29);(1,3)(2,8)(4,13)(5,14)(6,12)(7,19)(9,22)(10,21)(11,23)(15,26)(16,28)(17,27)(18,29)(20,30)(24,32)(25,31);(3,12)(8,21)(13,26)(14,27)(19,23)(22,31)(28,29)(30,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,e^2=g,g^2=h,b^a=bg,f^a=fh,e^b=eh,d^c=dh82 conjugacy classes — view the full character table.
| Fitting subgroup of | #33329⋊C3 |
| maximal subgroup of | #138086, #138087, #483009, … |
| maximal normal subgroup of | #138086, #138087, #483009, … |
| maximal quotient of | #479908, #479960, #479961, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26317);