Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#447 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
565 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 565 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,14)(4,18)(5,13)(6,12)(7,10)(8,15)(9,16)(11,21)(17,29)(19,30)(20,24)(22,28)(23,27)(25,31)(26,32);(1,3)(2,8)(4,19)(5,16)(6,23)(7,17)(9,24)(10,25)(11,27)(12,26)(13,30)(14,18)(15,21)(20,31)(22,32)(28,29);(1,3)(2,9)(4,15)(5,23)(6,16)(7,17)(8,24)(10,27)(11,25)(12,29)(13,14)(18,30)(19,21)(20,32)(22,31)(26,28);(2,10)(8,25)(9,27)(11,24)(12,13)(14,29)(18,28)(26,30)Polycyclic presentation:
2^8|e^2=gh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gh,e^b=eg,g^b=gh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #495739, #498124, #499746, … |
| maximal normal subgroup of | #495739, #498124, #499746, … |
| maximal quotient of | #484212, #484214, #484221, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26527);