Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9385 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
303 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,5,11,21,28,4,18)(3,13,17,26,31,32,15,27)(6,12,20,29,7,9,22,10)(8,16,25,23,24,14,30,19);(1,3)(2,8,12,26)(4,19)(5,23)(6,14)(7,16)(9,27,28,24)(10,25,18,32)(11,13,29,30)(15,20)(17,22)(21,31);(2,9)(3,14)(4,20)(5,22)(8,24)(12,28)(13,30)(15,17)(16,31)(19,23)(25,32)(26,27)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132831, #132832, #132835, … |
| maximal normal subgroup of | #132831, #132832, #132835, … |
| maximal quotient of | #100067, #100069, #100075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2399);