Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
249 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 249 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10)(11,19)(12,23,25,16)(13,26)(14,18)(15,21)(17,28)(20,33,36,40)(22,34)(24,31,27,32)(29,35,39,37)(30,38);(1,3,4,8,5,2,6,7)(9,11,13,21,14,22,28,38)(10,15,17,19,18,30,26,34)(12,20,27,37,25,36,24,35)(16,29,32,33,23,39,31,40);(9,12,14,25)(10,16,18,23)(11,20,22,36)(13,27,28,24)(15,29,30,39)(17,32,26,31)(19,33,34,40)(21,37,38,35)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,d^a=dg,e^a=eg,e^b=eg,d^c=dg,f^c=fg58 conjugacy classes — view the full character table.
| maximal subgroup of | #129586, #129587, #129588, … |
| maximal normal subgroup of | #129586, #129587, #129588, … |
| maximal quotient of | #99820, #99822, #99828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2088);