Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#546 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,8,12,9,11,15,18)(7,10,13,17,14,16,19,20);(1,2)(3,4)(5,7,9,14)(6,10,11,16)(8,13,15,19)(12,17,18,20);(6,11)(8,15)(10,16)(13,19);(2,4)Polycyclic presentation:
2^8|a^2=g,b^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,d^b=de,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #497288, #497290, #497294, … |
| maximal normal subgroup of | #497288, #497290, #497294, … |
| maximal quotient of | #219717, #219722, #219727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15396);