Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #124 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #397 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
273 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 273 conjugacy classes of subgroups — too many to draw. Only its 103 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:
(2,3)(5,6)(7,15,18,28)(8,12)(9,13)(10,14)(11,17,25,22)(16,31,19,30)(20,26)(21,27)(23,29)(24,34,35,33)(32,36);(1,2)(3,4)(5,7,9,22,10,18,23,17)(6,11,13,28,14,25,29,15)(8,19,20,33,21,16,32,34)(12,24,26,31,27,35,36,30);(1,3,4,2)(5,8,9,20,10,21,23,32)(6,12,13,26,14,27,29,36)(7,16,17,33,18,19,22,34)(11,24,15,30,25,35,28,31)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=d,d^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^b=dh,e^b=eh,g^b=gh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11320, #358369, #358423, … |
| maximal normal subgroup of | C2×#11320, #358369, #358423, … |
| maximal quotient of | #121086, #121087, #121088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4308);