Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
267 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 267 conjugacy classes of subgroups — too many to draw. Only its 85 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:
(1,2,4,7,5,3,6,8)(9,10)(11,19,21,23)(12,22,15,16)(13,17)(14,18)(20,24);(1,3,4,8,5,2,6,7)(9,11,13,22,14,21,24,16)(10,15,17,23,18,12,20,19);(9,12,14,15)(10,16,18,22)(11,20,21,17)(13,19,24,23)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,d^2=g,f^2=h,b^a=bd,c^a=ce,e^a=eg,d^b=dg,e^b=eg,f^b=fg,d^c=dg,f^c=fg64 conjugacy classes — view the full character table.
| maximal subgroup of | #94129, #94133, #94141, … |
| maximal normal subgroup of | #94129, #94133, #94141, … |
| maximal quotient of | #99536, #99548, #99549, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1946);