Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | Aut(G) order 65536 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
435 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 435 conjugacy classes of subgroups — too many to draw. Only its 139 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:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,21)(15,22,25,28)(16,20,26,27)(17,24,19,23);(5,7,8,11)(6,9,10,12)(13,15,17,16,18,25,19,26)(14,20,23,22,21,27,24,28);(6,10)(9,12)(14,21)(20,27)(22,28)(23,24);(2,4)(13,16,17,25,18,26,19,15)(14,22,23,27,21,28,24,20)Polycyclic presentation:
2^8|a^2=h,b^2=efh,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #125066, #125067, #125289, … |
| maximal normal subgroup of | #125066, #125067, #125289, … |
| maximal quotient of | #351334, #351335, #351338, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26124);