Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
509 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 509 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,9)(7,10,11,12)(13,14,16,20,18,22,17,21)(15,23,24,28,26,27,25,19);(1,2)(3,4)(6,9)(10,12)(13,15,18,26)(14,19,22,28)(16,25,17,24)(20,27,21,23);(2,4)(14,20)(16,17)(19,27)(21,22)(23,28)(24,25);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|a^2=e,b^2=h,d^2=ef,e^2=h,f^2=h,b^a=be,c^a=cf,c^b=cg,e^b=eh,f^b=fh,e^c=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #99704, #99706, #150330, … |
| maximal normal subgroup of | #99704, #99706, #150330, … |
| maximal quotient of | #244114, #244116, #244148, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16785);