Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13374 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #2023 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2023 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
180 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 180 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)(3,11,13,28)(4,16)(5,20)(6,9)(7,14,12,30)(8,18)(10,22)(15,24,21,29)(17,26,32,23)(19,27)(25,31);(1,2)(3,12,13,7)(4,8)(5,10)(6,9)(11,14,28,30)(15,26,21,23)(16,18)(17,29,32,24)(19,25)(20,22)(27,31);(1,3)(2,7)(4,17)(5,21)(6,13)(8,24)(9,12)(10,26)(11,27)(14,31)(15,22)(16,28)(18,30)(19,32)(20,23)(25,29)Polycyclic presentation:
2^8|a^2=d,b^2=d,e^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,f^b=fh,d^c=dh,e^c=egh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13374, #498976, #499269, … |
| maximal normal subgroup of | C2×#13374, #498976, #499269, … |
| maximal quotient of | #144454, #144455, #144484, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6362);