Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | Aut(G) order 16384 |
| 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 | C2×#562 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
469 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 469 conjugacy classes of subgroups — too many to draw. Only its 111 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)(3,11)(4,10)(5,9)(6,8)(7,12)(13,16)(14,15)(17,18,21,23)(19,25,27,32)(20,28,30,24)(22,29,31,26);(1,3)(2,7)(4,14)(5,12)(6,13)(8,15)(9,16)(10,11)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(20,30)(24,28)(25,32)(26,29);(2,8)(7,15)(9,10)(11,16)(18,23)(22,31)(24,28)(25,32)Polycyclic presentation:
2^8|a^2=h,b^2=h,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fgh,e^b=eg,f^b=fh,e^c=eh,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #245909, #245911, #245921, … |
| maximal normal subgroup of | #245909, #245911, #245921, … |
| maximal quotient of | #165888, #165920, #165956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11994);