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) = C25 |
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 | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
409 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 409 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(5,6)(7,16)(8,20,24,34)(9,25)(10,14)(11,15)(12,18)(13,23,31,22)(17,27,21,30)(19,28)(26,29)(32,33,36,35);(1,2)(3,4)(5,7)(6,12)(8,21)(9,26)(10,18)(11,19)(13,30)(14,28)(15,29)(16,25)(17,24)(20,32)(22,33)(23,35)(27,31)(34,36);(2,4)(5,8,9,22,11,24,10,23)(6,13,14,20,15,31,25,34)(7,17,18,33,19,21,26,35)(12,27,16,32,29,30,28,36)Polycyclic presentation:
2^8|a^2=g,c^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11150, #357909, #358423, … |
| maximal normal subgroup of | C2×#11150, #357909, #358423, … |
| maximal quotient of | #118666, #118698, #118702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4138);