Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| 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 | |
433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(5,6,7,8)(9,10,15,20)(11,21,22,24)(12,16,23,17)(13,19,14,18);(1,2)(3,4)(9,11)(10,16)(12,14)(13,23)(15,22)(17,20)(18,24)(19,21);(2,4)(9,12,13,22,15,23,14,11)(10,17,18,24,20,16,19,21)Polycyclic presentation:
2^8|a^2=g,c^2=d,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11192, #357963, #357964, … |
| maximal normal subgroup of | C2×#11192, #357963, #357964, … |
| maximal quotient of | #118692, #118811, #118815, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4180);