Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | Aut(G) = #72311213 (order 4096) · Inn(G) = #379 · Out(G) = C22×C22⋊C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #379 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1779 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
452 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 105 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:
(1,2,5,11,6,12,21,29,7,9,20,25,4,10,18,27)(3,13,15,30,14,23,31,28,16,26,19,32,17,8,22,24);(1,3)(2,8)(4,17)(5,19)(6,14)(7,16)(9,23)(10,26)(11,28)(12,13)(15,20)(18,31)(21,22)(24,25)(27,30)(29,32);(2,9)(5,20)(8,23)(10,12)(13,26)(15,19)(18,21)(22,31)Polycyclic presentation:
2^8|a^2=f,d^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dh,f^b=fh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #129238, #129239, #129273, … |
| maximal normal subgroup of | #129238, #129239, #129273, … |
| maximal quotient of | #129208, #129230, #129247, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5674);