Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×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 | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
485 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)(6,8)(9,10,14,17,13,18,12,15)(11,19,22,23,20,16,21,24);(2,4)(10,15,18,17)(11,20)(16,24,19,23)(21,22);(5,6,7,8)(9,11,12,21,13,20,14,22)(10,16,15,23,18,19,17,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10930, #354082, #354083, … |
| maximal normal subgroup of | C2×#10930, #354082, #354083, … |
| maximal quotient of | #110191, #110192, #110200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3918);