Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
589 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)(5,6,10,16,11,14,9,15)(7,12,19,26,18,27,24,30)(8,20,22,13,23,29,21,28)(17,31,33,35,32,36,34,25);(2,4)(5,7)(6,12,14,27)(8,17)(9,19)(10,24)(11,18)(13,25,28,35)(15,26,16,30)(20,31,29,36)(21,33)(22,34)(23,32);(5,8,11,23)(6,13,14,28)(7,17,18,32)(9,22,10,21)(12,25,27,35)(15,29,16,20)(19,34,24,33)(26,36,30,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=dh,f^b=fh,e^c=eh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10786, #353468, C4×#1905, … |
| maximal normal subgroup of | C2×#10786, #353468, C4×#1905, … |
| maximal quotient of | #105842, #105843, #105850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3774);