Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,15)(12,14)(13,20)(16,19)(17,23)(18,22)(21,24);(10,14)(11,17,18,21)(15,23,22,24)(16,20);(2,4)(9,11)(10,15)(12,18)(13,21)(14,22)(16,23)(17,19)(20,24)Polycyclic presentation:
2^8|a^2=g,b^2=dh,f^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^c=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11079, #357742, #357743, … |
| maximal normal subgroup of | C2×#11079, #357742, #357743, … |
| maximal quotient of | #115972, #115981, #115985, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4067);