Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C23 |
| commutator | C23 | C2×C42 |
| socle | C24 | C22×C4 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
767 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 767 conjugacy classes of subgroups — too many to draw. Only its 231 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,17,20,16,19,18,15);(2,4)(13,15,18,19,16,20,17,14);(6,8)(14,19)(15,20)Polycyclic presentation:
2^8|a^2=f,b^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | #92981, #92989, #93398, … |
| maximal normal subgroup of | #92981, #92989, #93398, … |
| maximal quotient of | #105481, #105482, #105491, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3703);