Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
297 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 297 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(6,7)(8,18)(9,13)(10,11)(14,15)(16,19)(17,20);(1,2)(3,4)(5,6)(7,18)(8,14)(9,20)(10,16)(11,15)(12,17)(13,19);(2,4)(5,7)(6,13)(8,17)(9,15)(10,14)(11,19)(12,20)(16,18)Polycyclic presentation:
2^8|d^2=gh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=eg,f^b=fh,g^b=gh,d^c=dgh,e^c=egh,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #125644, #125646, #125674, … |
| maximal normal subgroup of | #125644, #125646, #125674, … |
| maximal quotient of | #142640, #142641, #142720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6329);