Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
651 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 651 conjugacy classes of subgroups — too many to draw. Only its 223 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) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(9,10,11,12)(15,19)(17,20);(2,3)(6,8)(13,14,16,18)(15,17,19,20);(5,6,7,8)(13,15)(14,17)(16,19)(18,20);(1,3,4,2)(13,15)(14,17)(16,19)(18,20)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de64 conjugacy classes — view the full character table.
| maximal subgroup of | #101852, #101860, #101866, … |
| maximal normal subgroup of | #101852, #101860, #101866, … |
| maximal quotient of | #153961, #153982, #153996, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7412);