Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12727 |
| automorphism | Aut(G) = #72311574 (order 4096) · Inn(G) = #1782 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1782 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | #535 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #519 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
345 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 345 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:
(1,2,5,10,6,11,18,26)(3,12,15,27,14,8,22,24)(4,7,17,21,16,9,19,25)(13,23,29,32,28,20,30,31);(2,7)(4,16)(5,18)(8,20)(9,11)(10,21)(12,23)(13,28)(15,22)(24,31)(25,26)(27,32);(1,3)(2,8,11,12)(4,13)(5,15)(6,14)(7,20,9,23)(10,24,26,27)(16,28)(17,29)(18,22)(19,30)(21,31,25,32)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,f^b=fgh,e^c=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94450, #152078, #152079, … |
| maximal normal subgroup of | #94450, #152078, #152079, … |
| maximal quotient of | #129296, #129300, #129377, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5717);