Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | Aut(G) = #72303530 (order 4096) · Inn(G) = C2×#99 · Out(G) = #379 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#99 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C2×M16 | C23 |
| socle | C2 | C2×#324 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
182 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 182 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9,5,10,15,23,6,8,14,20,16,21,27,30)(3,7,12,18,13,19,26,29,11,17,24,22,25,28,32,31);(1,3)(2,7,8,17)(4,12)(5,13)(6,11)(9,22,20,18)(10,19,21,28)(14,24)(15,26)(16,25)(23,31,30,29)(27,32);(3,11)(7,17)(12,24)(13,25)(18,22)(19,28)(26,32)(29,31)Polycyclic presentation:
2^8|a^2=e,b^2=f,e^2=g,g^2=h,b^a=bd,d^a=df,f^a=fh,c^b=ch,d^b=dh,e^b=ef,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #129266, #129273, #129304, … |
| maximal normal subgroup of | #129266, #129273, #129304, … |
| maximal quotient of | #122314, #122315, #122318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4446);