Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
432 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 155 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:
(2,4)(5,6,8,10)(7,11,12,9)(13,14)(15,18)(16,19)(17,21)(20,24)(22,25)(23,26)(27,28);(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,23)(17,22)(19,26)(20,27)(21,25)(24,28);(1,2,3,4)(15,22)(18,25)(23,28)(26,27);(14,19)(16,24)(18,26)(20,21)(23,28)(25,27)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=f,g^2=h,b^a=be,c^a=cf,d^a=dg,g^a=gh,c^b=ch,g^d=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #273187, #273189, #273204, … |
| maximal normal subgroup of | #273187, #273189, #273204, … |
| maximal quotient of | #352532, #352534, #352536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26191);