Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| 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 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
358 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 117 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)(17,18,21,24)(19,25,27,32)(20,29,26,23)(22,30,31,28);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16);(10,13)(14,16)(17,19,26,30,21,27,20,28)(18,22,29,32,24,31,23,25);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,27)(22,31)(25,32)(28,30)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=fh,d^2=g,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #174986, #174988, #175010, … |
| maximal normal subgroup of | #174986, #174988, #175010, … |
| maximal quotient of | #363755, #364075, #371519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22489);