Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #389 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6,9,15,11,16,10,14)(7,17,20,28,21,27,19,12)(8,22,23,13,25,29,24,30)(18,26,33,31,34,36,32,35);(5,7,10,19,11,21,9,20)(6,12,14,27,16,28,15,17)(8,18,24,32,25,34,23,33)(13,26,22,35,30,36,29,31);(5,8,9,23,11,25,10,24)(6,13,14,22,16,30,15,29)(7,18,20,33,21,34,19,32)(12,26,27,35,28,36,17,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=e,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #146758, #146761, #146765, … |
| maximal normal subgroup of | #146758, #146761, #146765, … |
| maximal quotient of | #108900, #108901, #108904, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3861);