Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1215 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 1215 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(5,6,7,8)(10,12);(6,8)(9,10)(11,12)(14,16);(5,6,7,8)(9,10,11,12)(13,14)(15,16);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)Polycyclic presentation:
2^8|a^2=ef,c^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de46 conjugacy classes — view the full character table.
| Fitting subgroup of | #17091⋊C3 |
| maximal subgroup of | C2×#17091, #844862, #844869, … |
| maximal normal subgroup of | C2×#17091, #844862, #844869, … |
| maximal quotient of | #156467, #156468, #156474, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10079);