Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8528 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
364 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 75 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:
(1,2,4,6,5,3,7,8)(9,10,14,16,13,12,20,21)(11,15,19,23,18,17,22,24);(1,3)(2,5)(4,6)(7,8)(9,11,13,18)(10,15,12,17)(14,22,20,19)(16,24,21,23);(4,7)(6,8)(9,12)(10,13)(11,17)(14,16)(15,18)(19,23)(20,21)(22,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,e^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,e^c=eg,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152566, #152571, #153134, … |
| maximal normal subgroup of | #152566, #152571, #153134, … |
| maximal quotient of | #97045, #97049, #97261, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1519);