Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
153 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 153 conjugacy classes of subgroups — too many to draw. Only its 55 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,12,32,13,19,30,18)(7,20,25,33,8,26,28,35)(9,29,21,16,10,17,11,31)(14,27,36,23,15,24,34,22);(1,2)(3,4)(5,7,21,27,12,25,10,23,13,8,11,24,30,28,9,22)(6,14,29,35,18,34,31,26,19,15,17,33,32,36,16,20);(2,4)(5,8,13,7)(6,15,19,14)(9,24,10,27)(11,22,21,23)(12,25,30,28)(16,33,31,35)(17,20,29,26)(18,34,32,36)Polycyclic presentation:
2^8|a^2=de,b^2=dg,c^2=h,d^2=gh,e^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gh,c^b=cf,d^c=dg,e^c=eg,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13471, #499313, #499315, … |
| maximal normal subgroup of | C2×#13471, #499313, #499315, … |
| maximal quotient of | #145905, #145915, #145925, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6459);