Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| 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 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#576 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,14,10,7,15,13)(8,17,19,16,20,12,18,11);(5,7)(6,10)(8,16)(9,13)(11,20)(12,19)(14,15)(17,18);(2,4)(5,8)(6,11)(7,16)(9,18)(10,20)(12,14)(13,17)(15,19);(5,8)(6,12)(7,17)(9,19)(10,20)(11,14)(13,16)(15,18)Polycyclic presentation:
2^8|a^2=g,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,d^b=de,e^b=eh,f^c=fh,g^c=gh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #497459, #497460, #497462, … |
| maximal normal subgroup of | #497459, #497460, #497462, … |
| maximal quotient of | #220223, #220229, #220237, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15454);