Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18562 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3519 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
504 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 504 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,9,10,16)(4,12,14,7)(6,13,15,11)(17,18)(19,27)(20,25)(21,24)(22,26)(23,29)(28,31)(30,32);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,19)(18,23)(20,30)(21,29)(22,28)(24,31)(25,27)(26,32);(18,24)(20,21)(23,31)(25,26)(27,32)(29,30);(3,10)(6,15)(9,16)(11,13)(19,28)(23,32)(27,31)(29,30)Polycyclic presentation:
2^8|a^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #273187, #273189, #273211, … |
| maximal normal subgroup of | #273187, #273189, #273211, … |
| maximal quotient of | #166298, #166346, #166426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11559);