Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| prime radical | #3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,18,24)(12,20,21,15)(14,22,23,19);(2,5)(6,8);(9,11,12,19,13,18,21,22)(10,14,15,17,16,23,20,24);(1,3,4,7)(2,6,5,8);(11,18)(14,23)(17,24)(19,22)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=f,g^2=h,b^a=bf,c^a=cg,g^a=gh,e^c=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #112722, #112724, #112748, … |
| maximal normal subgroup of | #112722, #112724, #112748, … |
| maximal quotient of | #6371209, #6372124, #6372439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53689);