Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
329 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,13,15,16)(17,18,21,24)(19,25,27,23)(20,22,30,31)(26,28,29,32);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(17,19,21,27)(18,23,24,25)(20,29,30,26)(22,28,31,32);(11,15)(13,16)(17,20,22,24,21,30,31,18)(19,26,28,23,27,29,32,25);(1,3,4,8)(2,6,5,7)(18,24)(19,27)(20,30)(28,32)Polycyclic presentation:
2^8|a^2=egh,b^2=g,c^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #150017, #150019, #153076, … |
| maximal normal subgroup of | #150017, #150019, #153076, … |
| maximal quotient of | #362188, #363453, #365568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22193);