Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
519 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 519 conjugacy classes of subgroups — too many to draw. Only its 135 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)(13,14,16,20,18,21,17,22)(15,23,24,28,26,27,25,19);(1,2)(3,4)(6,9)(10,12)(13,15,18,26)(14,19,21,28)(16,25,17,24)(20,27,22,23);(2,4)(14,20)(16,17)(19,27)(21,22)(23,28)(24,25);(5,7,8,11)(6,10,9,12)(14,21)(19,28)(20,22)(23,27)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,d^a=dh,c^b=cg,e^b=eh,f^b=fh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #232171, #232177, #232183, … |
| maximal normal subgroup of | #232171, #232177, #232183, … |
| maximal quotient of | #244124, #244161, #244162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16792);