Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | #3169 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
189 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 189 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,3,8,6)(9,10)(11,14)(12,21)(13,17)(15,18)(16,19)(20,23)(22,24);(1,3)(2,5)(4,6)(7,8)(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22);(2,3)(6,7)Polycyclic presentation:
2^8|a^2=e,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=cg,d^b=df,f^b=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #130427, #130429, #130435, … |
| maximal normal subgroup of | #130427, #130429, #130435, … |
| maximal quotient of | #125094, #125114, #125129, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4659);