Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | Aut(G) = C2×#72236881 (order 2048) · Inn(G) = #555 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2420 |
| prime radical | #3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
323 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 105 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)(3,15)(4,19)(5,9)(6,13)(7,12)(8,14)(10,22)(11,18)(16,31)(17,26)(20,28)(21,27)(23,29)(24,30)(25,32);(1,3,7,17,8,18,6,5)(2,9,13,11,14,26,12,15)(4,20,22,16,24,21,23,32)(10,27,19,31,29,28,30,25);(1,4)(2,10)(3,16)(5,21)(6,22)(7,23)(8,24)(9,25)(11,28)(12,19)(13,30)(14,29)(15,27)(17,20)(18,32)(26,31);(1,5,6,18,8,17,7,3)(2,11,12,9,14,15,13,26)(4,21,22,32,24,20,23,16)(10,28,19,25,29,27,30,31)Polycyclic presentation:
2^8|b^2=f,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #501856, #501857, #501994, … |
| maximal normal subgroup of | #501856, #501857, #501994, … |
| maximal quotient of | #280843, #280844, #280845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17973);