Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = C25×#442 (order 2048) · Inn(G) = C2×D8 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C4×D8 |
| prime radical | C2×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
238 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(2,4)(5,6,11,18,12,19,9,16,13,20,29,35,21,30,10,17)(7,15,23,14,24,32,8,26,25,31,28,33,27,36,22,34);(1,2,3,4)(5,7,21,24,13,25,12,27)(6,14,30,26,20,33,19,34)(8,10,28,29,22,9,23,11)(15,16,32,18,31,17,36,35);(5,8)(6,15)(7,10)(9,25)(11,27)(12,23)(13,22)(14,16)(17,33)(18,26)(19,36)(20,31)(21,28)(24,29)(30,32)(34,35)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,e^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,d^b=dg,e^b=eg,f^b=fg,g^b=gh,d^c=dg,e^c=eg,f^c=fg,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13011, #494490, #494491, … |
| maximal normal subgroup of | C2×#13011, #494490, #494491, … |
| maximal quotient of | #136685, #136686, #136687, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5999);