Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 32768 |
| 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 | |
643 conjugacy classes of subgroups; only the 203 normal subgroups are shown.
This group has 643 conjugacy classes of subgroups — too many to draw. Only its 203 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)(6,8)(13,14,17,19,16,15,18,20);(2,4)(9,10,11,12)(14,15)(19,20);(5,6,7,8)(13,15,17,20,16,14,18,19)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,f^2=h,b^a=bd,c^a=ce,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | #93546, #93572, #97105, … |
| maximal normal subgroup of | #93546, #93572, #97105, … |
| maximal quotient of | #110043, #110046, #110047, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3910);