Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #102 |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(13,14,17,21)(15,16,23,25)(18,20,26,28)(19,27,24,22);(6,9)(7,12)(10,11)(13,15,18,22,19,20,25,21,17,23,26,27,24,28,16,14)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,e^2=fgh,f^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7275, #127708, #127709, … |
| maximal normal subgroup of | C2×#7275, #127708, #127709, … |
| maximal quotient of | #93063, #93065, #93070, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 263);