Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13391 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #2033 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2033 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
195 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 195 conjugacy classes of subgroups — too many to draw. Only its 53 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)(3,14)(4,18)(5,20)(6,22)(7,11)(8,13)(9,15)(10,19)(12,21)(16,26)(17,27)(23,25)(24,28)(29,32)(30,31);(1,3)(2,9)(4,5)(6,16)(7,15)(8,17)(10,28)(11,27)(12,26)(13,18)(14,20)(19,21)(22,30)(23,31)(24,32)(25,29);(1,4,5,3,8,15,7,17)(2,10,11,29,13,30,20,26)(6,16,21,31,24,32,23,19)(9,25,14,28,18,12,27,22)Polycyclic presentation:
2^8|c^2=d,d^2=h,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,c^b=cf,d^b=dh,f^b=fh,e^c=egh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13391, #499027, #499514, … |
| maximal normal subgroup of | C2×#13391, #499027, #499514, … |
| maximal quotient of | #144688, #144689, #144780, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6379);