Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | #3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
715 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 715 conjugacy classes of subgroups — too many to draw. Only its 211 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) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,15,13,16,11,14);(10,14,16,15);(2,5)(6,8);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh88 conjugacy classes — view the full character table.
| Frattini subgroup of | #72272677 |
| maximal subgroup of | #107759, #108079, #108093, … |
| maximal normal subgroup of | #107759, #108079, #108093, … |
| maximal quotient of | #207540, #207541, #207542, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14403);