Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
149 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,17,22,15,23,24,28,16,21,25,27,18,19,26,20);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,16,18)(14,19,21,23)(17,24,25,26)(20,27,28,22);(2,4)(5,8,9,7)(6,10,11,12)(14,20)(15,18)(17,26)(19,22)(21,28)(23,27)(24,25)Polycyclic presentation:
2^8|a^2=e,b^2=d,c^2=dh,e^2=gh,g^2=h,b^a=bd,c^a=ce,c^b=cf,e^c=eg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93856, #93858, #151499, … |
| maximal normal subgroup of | #93856, #93858, #151499, … |
| maximal quotient of | #146144, #146155, #146173, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6503);