Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9383 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
253 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 253 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,6)(7,8)(9,10,15,18,14,19,13,17)(11,20,22,16,23,24,21,12);(1,2,6,8,4,7,3,5)(9,11)(10,16)(12,19)(13,22)(14,23)(15,21)(17,24)(18,20);(9,12,14,16)(10,11,19,23)(13,20,15,24)(17,22,18,21)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=h,d^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,f^c=fh49 conjugacy classes — view the full character table.
| maximal subgroup of | #130213, #130214, #130215, … |
| maximal normal subgroup of | #130213, #130214, #130215, … |
| maximal quotient of | #100062, #100078, #100750, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2382);