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:
(1,2,3,4)(5,6)(7,16)(8,13)(9,24)(10,15)(11,14)(12,18)(17,32)(19,33)(20,27)(21,26)(22,30)(23,31)(25,29)(28,34)(35,36);(5,7,11,20,10,19,9,18)(6,12,24,33,15,27,14,16)(8,21,22,34,23,35,25,17)(13,28,29,26,31,32,30,36);(5,8,9,22,10,23,11,25)(6,13,14,29,15,31,24,30)(7,17,18,21,19,34,20,35)(12,26,16,32,27,36,33,28)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=d,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11305, #358342, #358420, … |
| maximal normal subgroup of | C2×#11305, #358342, #358420, … |
| maximal quotient of | #121079, #121095, #121097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4293);