Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C23×#41 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2195 conjugacy classes of subgroups; only the 351 normal subgroups are shown.
This group has 2195 conjugacy classes of subgroups — too many to draw. Only its 351 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,8)(13,14)(15,17)(16,18)(19,20);(10,12)(14,15)(18,19);(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,15,17,14)(16,18,20,19);(13,16,17,20)(14,18,15,19)Polycyclic presentation:
2^8|c^2=h,d^2=e,b^a=be,c^a=cf,c^b=cg,d^c=de64 conjugacy classes — view the full character table.
| Fitting subgroup of | #14337⋊C3 |
| maximal subgroup of | #107117, #108243, #115233, … |
| maximal normal subgroup of | #107117, #108243, #115233, … |
| maximal quotient of | #153933, #153944, #153945, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7325);