Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | Aut(G) = #72286320 (order 2048) · Inn(G) = #353 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #353 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #124 | C23 |
| socle | C2 | C2×#353 |
| prime radical | #3150 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
204 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 45 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)(3,13,15,29)(4,9)(5,19)(6,11,22,26)(7,12)(8,14,24,31)(10,21,27,18)(16,23,17,30)(20,28,32,25);(1,3)(2,8)(4,14)(5,20)(6,15)(7,17)(9,23)(10,13)(11,28)(12,25)(16,22)(18,31)(19,29)(21,32)(24,26)(27,30);(1,4,7,5)(2,9,12,19)(3,14,17,20)(6,18,22,21)(8,23,25,29)(10,26,27,11)(13,24,30,28)(15,31,16,32)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=g,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,e^d=egh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #151921, #151922, #151956, … |
| maximal normal subgroup of | #151921, #151922, #151956, … |
| maximal quotient of | #149567, #149628, #149685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6565);