Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #389 |
| prime radical | C23 | #125 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8,6,9,4,13)(3,10,11,7,12,15,14,16)(18,23)(19,26)(20,21)(24,31)(25,29)(27,28);(1,3,5,11,6,12,4,14)(2,7,8,15,9,16,13,10)(17,18)(19,24)(20,29)(21,23)(22,25)(26,30)(27,32)(28,31);(17,19,21,26,22,28,20,27)(18,24,23,30,25,31,29,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=e,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dg58 conjugacy classes — view the full character table.
| maximal subgroup of | #136484, #136485, #136571, … |
| maximal normal subgroup of | #136484, #136485, #136571, … |
| maximal quotient of | #101927, #101928, #101932, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2621);