Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C22×C42 |
| socle | C23 | C23×C4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1081 conjugacy classes of subgroups; only the 571 normal subgroups are shown.
This group has 1081 conjugacy classes of subgroups — too many to draw. Only its 571 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(2,4)(5,6,7,8);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,f^2=h,b^a=be,d^c=dh100 conjugacy classes — view the full character table.
| Fitting subgroup of | #20336⋊C3 |
| maximal subgroup of | #100523, #100527, #100531, … |
| maximal normal subgroup of | #100523, #100527, #100531, … |
| maximal quotient of | #180849, #180853, #180856, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13324);