Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2657 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
502 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 502 conjugacy classes of subgroups — too many to draw. Only its 143 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)(7,10)(8,9)(11,12)(14,18)(15,21)(16,23)(19,24)(20,26)(22,27);(6,9)(10,12)(13,14,16,18,17,19,23,24)(15,20,21,25,22,26,27,28);(1,2,3,4)(14,19)(18,24)(20,26)(25,28);(5,7,8,11)(6,10,9,12)(13,15,16,21,17,22,23,27)(14,20,18,25,19,26,24,28)Polycyclic presentation:
2^8|b^2=g,c^2=f,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #300422, #300423, #300425, … |
| maximal normal subgroup of | #300422, #300423, #300425, … |
| maximal quotient of | #352625, #352627, #352629, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26216);