Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
309 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(7,9)(8,11)(10,12)(13,14,19,16)(15,23,27,24)(17,22,18,21)(20,26,28,25);(2,4)(5,6)(7,9)(8,11)(10,12)(13,15,19,27)(14,20,16,28)(17,26,18,25)(21,24,22,23);(6,9)(7,12)(10,11)(13,16,19,14)(15,24,27,23)(17,21,18,22)(20,25,28,26)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=eh,f^b=fgh,d^c=dg,e^c=eh,f^c=fgh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94645, #125650, #125724, … |
| maximal normal subgroup of | #94645, #125650, #125724, … |
| maximal quotient of | #102792, #102798, #102855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3297);