Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13417 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2041 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3155 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
227 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 227 conjugacy classes of subgroups — too many to draw. Only its 55 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 — 3 generators on 32 points:
(1,2)(3,15)(4,9)(5,21)(6,12)(7,11)(8,14)(10,18)(13,23)(16,26)(17,29)(19,27)(20,31)(22,28)(24,30)(25,32);(2,9)(3,16)(4,7)(5,6)(10,27)(11,14)(12,31)(15,30)(17,18)(19,22)(20,23)(24,25)(26,29)(28,32);(1,3,4,17,8,19,7,25)(2,10,11,28,14,30,9,26)(5,22,20,18,23,16,6,24)(12,29,13,15,31,32,21,27)Polycyclic presentation:
2^8|c^2=d,d^2=h,e^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,f^a=fg,g^a=gh,c^b=cf,d^b=dh,e^b=eg,f^b=fgh,g^b=gh,e^c=egh,f^c=fg,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13417, #499091, #499269, … |
| maximal normal subgroup of | C2×#13417, #499091, #499269, … |
| maximal quotient of | #144895, #144896, #144922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6405);