Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | #3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,12,16,13,15,19,22)(11,14,18,20,17,21,23,24);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(10,15)(11,17)(16,22)(18,23);(1,3,5,4)(2,6,7,8)Polycyclic presentation:
2^8|a^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,c^b=ch,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #107293, #107294, #107740, … |
| maximal normal subgroup of | #107293, #107294, #107740, … |
| maximal quotient of | #161916, #161917, #161918, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10469);