Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
623 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,8,5,6,3,7)(9,10)(11,19,22,15)(12,16)(13,17)(14,18)(20,24,21,23);(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,19,24);(1,3,5,4)(2,7,6,8)(9,13,14,12)(10,17,18,16)(11,21,22,20)(15,24,19,23)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=gh,f^2=h,b^a=be,c^a=ch,e^a=eg,f^b=fg64 conjugacy classes — view the full character table.
| maximal subgroup of | #105584, #105591, #105620, … |
| maximal normal subgroup of | #105584, #105591, #105620, … |
| maximal quotient of | #160713, #160719, #160724, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10354);