Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 32768 |
| 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 | C2×C4⋊C4 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
397 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6,7,8)(9,10,12,17,14,18,13,16)(11,19,21,24,22,23,20,15);(9,11,13,20,14,22,12,21)(10,15,16,23,18,24,17,19);(1,2,3,4)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=e,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #106951, #107072, #107073, … |
| maximal normal subgroup of | #106951, #107072, #107073, … |
| maximal quotient of | #108955, #108956, #108960, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3841);