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:
(5,6,10,16,11,14,9,15)(7,17,19,29,21,12,20,28)(8,22,24,31,26,32,25,13)(18,34,35,33,36,30,23,27);(1,2)(3,4)(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,30)(14,17)(15,28)(16,29)(18,25)(22,33)(24,36)(26,35)(27,32)(31,34);(2,4)(5,8)(6,13,14,31)(7,18)(9,24)(10,25)(11,26)(12,27,17,33)(15,22,16,32)(19,23)(20,35)(21,36)(28,30,29,34)Polycyclic presentation:
2^8|a^2=g,c^2=d,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fh,g^b=gh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11177, #357930, #358420, … |
| maximal normal subgroup of | C2×#11177, #357930, #358420, … |
| maximal quotient of | #118682, #118760, #118763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4165);