Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18376 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3490 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
541 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 541 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(1,2,5,10)(3,12,13,25)(4,16,19,9)(6,11,20,8)(7,17,22,14)(15,27,28,32)(18,24,31,26)(21,30,23,29);(1,3)(2,7)(4,17)(5,13)(6,15)(8,23)(9,25)(10,22)(11,21)(12,16)(14,19)(18,30)(20,28)(24,27)(26,32)(29,31);(4,18)(9,26)(12,27)(14,29)(16,24)(17,30)(19,31)(25,32);(2,8)(3,13)(7,21)(9,24)(10,11)(12,27)(14,17)(15,28)(16,26)(22,23)(25,32)(29,30)Polycyclic presentation:
2^8|a^2=g,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,d^b=dg,e^b=eg,f^b=fh,e^c=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | #486099, #486101, #486109, … |
| maximal normal subgroup of | #486099, #486101, #486109, … |
| maximal quotient of | #169614, #169626, #169678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11365);