Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
211 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 211 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,11,17,12,19,31,18)(7,20,24,33,26,8,25,28)(9,29,10,15,21,16,32,36)(13,14,34,22,35,23,27,30);(1,2,3,4)(5,7,21,22,11,25,32,14,12,26,9,30,31,24,10,23)(6,13,15,8,18,34,29,28,19,35,36,20,17,27,16,33);(5,8,9,27,11,28,10,13,12,20,21,34,31,33,32,35)(6,14,36,7,18,30,16,25,19,23,15,26,17,22,29,24)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=egh,d^2=g,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,d^b=dh,f^b=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13246, #495232, #495233, … |
| maximal normal subgroup of | C2×#13246, #495232, #495233, … |
| maximal quotient of | #140730, #140772, #141209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6234);