Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 8192 |
| 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 | |
335 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)(7,17,19,12,21,27,20,28)(8,22,23,30,25,29,24,13)(18,26,32,35,34,36,33,31);(2,4)(5,7,10,20,11,21,9,19)(6,12,15,27,16,28,14,17)(8,18,24,33,25,34,23,32)(13,26,22,35,30,36,29,31);(5,8,9,23,11,25,10,24)(6,13,15,22,16,30,14,29)(7,18,19,32,21,34,20,33)(12,26,27,35,28,36,17,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,e^c=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10898, #353924, #354318, … |
| maximal normal subgroup of | C2×#10898, #353924, #354318, … |
| maximal quotient of | #109512, #109513, #109516, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3886);