Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8101 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#95 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9,6,8,5,10)(3,11,12,16,13,7,14,15)(17,18,21,25,22,26,20,24)(19,27,30,31,29,23,28,32);(1,3,5,12,6,13,4,14)(2,7,10,15,8,11,9,16)(17,19)(18,23,26,27)(20,28)(21,30)(22,29)(24,31,25,32);(2,8)(7,11)(9,10)(15,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,27,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,e^2=gh,f^2=g,b^a=bd,c^a=cg,d^a=dg,f^a=fh,d^b=dh,e^b=eg52 conjugacy classes — view the full character table.
| Fitting subgroup of | #8126⋊C3 |
| maximal subgroup of | #105889, #105890, #105938, … |
| maximal normal subgroup of | #105889, #105890, #105938, … |
| maximal quotient of | #95352, #95357, #95366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1114);