Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
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 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
297 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 297 conjugacy classes of subgroups — too many to draw. Only its 135 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,15,16,13)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,32)(28,31);(1,3,5,4)(2,6,8,7)(17,19,25,26,21,28,20,27)(18,22,29,30,23,31,24,32);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(19,26)(20,25)(22,30)(24,29)(27,28)(31,32);(1,4,5,3)(2,7,8,6)(18,23)(22,31)(24,29)(30,32)Polycyclic presentation:
2^8|a^2=f,b^2=egh,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #144551, #144560, #145890, … |
| maximal normal subgroup of | #144551, #144560, #145890, … |
| maximal quotient of | #362245, #362275, #365851, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21822);