Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C4×D4 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
391 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)(9,10,13,17,14,18,12,15)(11,16,20,23,21,24,19,22);(2,4)(10,15,18,17)(16,22,24,23);(6,8)(9,11,13,20,14,21,12,19)(10,16,17,23,18,24,15,22)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10882, #353884, #353885, … |
| maximal normal subgroup of | C2×#10882, #353884, #353885, … |
| maximal quotient of | #109467, #109469, #109470, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3870);