Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
465 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 465 conjugacy classes of subgroups — too many to draw. Only its 107 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:
(1,2,3,5,4,6,7,8)(9,10)(11,19)(12,23,25,16)(13,26)(14,18)(15,21)(17,28)(20,34,36,33)(22,32)(24,31,40,38)(27,30)(29,35,37,39);(9,11,14,22)(10,15,18,30)(12,20,25,36)(13,27,28,21)(16,29,23,37)(17,32,26,19)(24,39,40,35)(31,34,38,33);(9,12)(10,16)(11,20)(13,24)(14,25)(15,29)(17,31)(18,23)(19,33)(21,35)(22,36)(26,38)(27,39)(28,40)(30,37)(32,34)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,f^c=fg58 conjugacy classes — view the full character table.
| maximal subgroup of | #129233, #129235, #129272, … |
| maximal normal subgroup of | #129233, #129235, #129272, … |
| maximal quotient of | #96982, #96984, #96987, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1311);