Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
311 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 99 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:
(2,4)(6,9)(7,10)(11,12)(13,14,17,18,16,19,15,20);(5,6,10,12,8,11,7,9)(14,18)(15,17)(19,20);(1,2,3,4)(14,19)(18,20)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=eh,d^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #98006, #98007, #98008, … |
| maximal normal subgroup of | #98006, #98007, #98008, … |
| maximal quotient of | #101958, #101972, #102220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2686);