Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
847 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 847 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)(9,10,11,14,12,13,15,16);(2,5)(6,8);(10,13)(11,15);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=g,d^2=e,g^2=h,b^a=be,c^a=cf,f^a=fh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #105582, #105589, #105618, … |
| maximal normal subgroup of | #105582, #105589, #105618, … |
| maximal quotient of | #190446, #190447, #190454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13714);