Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2035 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 2035 conjugacy classes of subgroups — too many to draw. Only its 185 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2)(3,4)(6,9)(7,11)(10,12);(5,6)(7,12)(8,10)(9,11);(7,11)(9,12);(2,4)Polycyclic presentation:
2^8|e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,e^b=eh,f^b=fh,e^c=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | #248647, #248649, #248653, … |
| maximal normal subgroup of | #248647, #248649, #248653, … |
| maximal quotient of | #346294, #346295, #346300, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25876);