Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C4×D4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
443 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)(9,10,13,18,15,11,14,17)(12,20,22,19,24,21,23,16);(5,6)(7,8)(9,11)(10,15)(12,21)(13,17)(14,18)(16,22)(19,23)(20,24);(2,4)(6,8)(9,12,13,22,15,24,14,23)(10,16,17,21,11,19,18,20)Polycyclic presentation:
2^8|a^2=g,c^2=d,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11191, #357959, #357960, … |
| maximal normal subgroup of | C2×#11191, #357959, #357960, … |
| maximal quotient of | #118690, #118691, #118783, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4179);