Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C2×#459 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
348 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 117 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)(3,12)(4,17)(5,11)(6,21)(7,22)(8,13)(9,15)(10,14)(16,29)(18,32)(19,28)(20,24)(23,25)(26,30)(27,31);(1,3,6,15,8,16,5,7)(2,9,11,12,13,22,21,29)(4,14,19,30,20,31,18,23)(10,24,25,28,27,17,26,32);(1,4)(2,10)(3,14)(5,18)(6,19)(7,23)(8,20)(9,24)(11,25)(12,28)(13,27)(15,30)(16,31)(17,22)(21,26)(29,32);(1,5,8,6)(2,11,13,21)(3,7,16,15)(4,18,20,19)(9,12,22,29)(10,25,27,26)(14,23,31,30)(17,32,24,28)Polycyclic presentation:
2^8|b^2=e,d^2=h,e^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fg,g^a=gh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136867, #136870, #136873, … |
| maximal normal subgroup of | #136867, #136870, #136873, … |
| maximal quotient of | #490427, #490428, #490429, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26582);