Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = C2×#72237167 (order 4096) · Inn(G) = C2×D8 · Out(G) = C2×#525 |
| 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 | C2×#102 |
| prime radical | #3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
259 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 259 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) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,14,10,15,24,32,11,16,25,33,23,31,8,13)(7,17,19,29,20,30,18,34,21,35,22,12,26,27,36,28);(5,7,10,26,11,21,23,20)(6,12,15,34,16,29,31,28)(8,22,9,18,24,19,25,36)(13,30,14,17,32,27,33,35);(2,4)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,d^b=dg,f^b=fg,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #131272, #131279, #131330, … |
| maximal normal subgroup of | #131272, #131279, #131330, … |
| maximal quotient of | #136603, #136604, #136605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5978);