Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #390 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1439 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
426 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 117 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)(3,12)(4,16,18,26)(5,19)(6,10)(7,11)(8,14)(9,17,27,21)(13,25,31,22)(15,23)(20,24)(28,30,32,29);(1,3,5,14,7,15,6,20)(2,8,10,12,11,24,19,23)(4,13,17,29,18,31,21,30)(9,25,26,32,27,22,16,28);(1,4)(2,9)(3,13)(5,17)(6,21)(7,18)(8,22)(10,26)(11,27)(12,28)(14,29)(15,31)(16,19)(20,30)(23,32)(24,25)Polycyclic presentation:
2^8|a^2=g,b^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=eh,e^c=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131356, #131357, #144728, … |
| maximal normal subgroup of | #131356, #131357, #144728, … |
| maximal quotient of | #116433, #116463, #116465, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4097);