Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×D8 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C2 | C2×D4⋊C8 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
190 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 190 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,11,21,26,7,12,22,27,20,10,4,18)(3,13,15,8,17,24,31,25,14,29,30,23,16,32,19,28);(1,3)(2,8)(4,19)(5,15)(6,17)(7,14)(9,24)(10,28)(11,25)(12,23)(13,18)(16,20)(21,31)(22,30)(26,29)(27,32);(3,14)(8,23)(13,29)(15,30)(16,17)(19,31)(24,32)(25,28)Polycyclic presentation:
2^8|a^2=e,d^2=fh,e^2=g,f^2=h,g^2=h,b^a=bd,d^a=df,f^a=fh,c^b=ch,d^b=df,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #128554, #128583, #128695, … |
| maximal normal subgroup of | #128554, #128583, #128695, … |
| maximal quotient of | #122441, #122442, #122443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4590);