Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8746 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #381 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,16,13,14,20,21)(11,17,19,24,18,15,22,23);(10,14)(12,20)(15,17)(19,22);(1,3,4,8)(2,5,6,7)(9,11)(10,15,14,17)(12,19)(13,18)(16,23,21,24)(20,22)Polycyclic presentation:
2^8|a^2=f,c^2=dh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fg,e^c=eg,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #133486, #133487, #133488, … |
| maximal normal subgroup of | #133486, #133487, #133488, … |
| maximal quotient of | #99145, #99152, #99154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1740);