Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
195 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,16)(17,21)(18,28)(19,24)(20,23)(22,25)(26,27);(2,4)(6,9)(10,11)(13,15,25,24,18,26,27,28,19,22,16,14,20,21,17,23);(5,7,8,12)(6,10,9,11)(13,16,18,17,19,25,20,27)(14,21,23,15,24,26,28,22)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=e,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #131917, #131918, #131947, … |
| maximal normal subgroup of | #131917, #131918, #131947, … |
| maximal quotient of | #141885, #141912, #142002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6271);