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×D4 | C23 |
| prime radical | #535 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
357 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 357 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)(3,11,14,24)(4,15)(5,18,20,9)(6,10)(7,13)(8,17)(12,26,28,25)(16,23,32,30)(19,22)(21,31)(27,29);(1,3,6,14)(2,7,10,22)(4,12,17,28)(5,19,20,13)(8,21,15,29)(9,24,18,11)(16,31,32,27)(23,26,30,25);(1,4)(2,8,10,15)(3,12)(5,16)(6,17)(7,21,22,29)(9,23,18,30)(11,25,24,26)(13,27)(14,28)(19,31)(20,32)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=g,d^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 | #94451, #152078, #152079, … |
| maximal normal subgroup of | #94451, #152078, #152079, … |
| maximal quotient of | #129297, #129300, #129378, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5716);