Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
348 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 145 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,26,21,23,20,25)(19,27,29,32,28,24,30,31);(1,2)(3,8)(4,5)(6,7)(11,16)(13,15)(18,23)(19,28)(25,26)(29,30);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19,21,28)(18,24,23,27)(20,30,22,29)(25,32,26,31);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,21,22)(18,25,23,26)(19,29,28,30)(24,31,27,32)Polycyclic presentation:
2^8|a^2=g,c^2=f,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cefh,d^b=deh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #136299, #137299, #139960, … |
| maximal normal subgroup of | #136299, #137299, #139960, … |
| maximal quotient of | #357305, #357339, #357345, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16339);