Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = C22×#72237160 (order 4096) · Inn(G) = C2×D8 · Out(G) = C23×#41 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
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 | C2×#107 |
| prime radical | C23 | C2×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
143 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 143 conjugacy classes of subgroups — too many to draw. Only its 77 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,9,14,11,15,10,25)(7,16,18,32,19,33,26,30)(8,20,22,31,24,35,23,13)(12,21,27,34,29,36,28,17);(5,7,9,18,11,19,10,26)(6,12,14,27,15,29,25,28)(8,21,22,34,24,36,23,17)(13,30,20,16,31,32,35,33);(1,2,3,4)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=e,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #136333, #136336, #136349, … |
| maximal normal subgroup of | #136333, #136336, #136349, … |
| maximal quotient of | #140377, #140378, #140379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6167);