Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#120 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,15)(8,17)(9,12)(10,13)(11,14)(16,19)(18,20);(1,3,4,2)(5,7)(6,8)(9,16)(10,17)(11,18)(12,15)(13,20)(14,19);(5,8)(6,7)(9,15)(10,20)(11,19)(12,16)(13,17)(14,18);(5,9,11,10)(6,12,14,13)(7,16,18,17)(8,15,19,20)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #354483, #354484, #356752, … |
| maximal normal subgroup of | #354483, #354484, #356752, … |
| maximal quotient of | #191808, #191813, #191818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13851);