Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
533 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 533 conjugacy classes of subgroups — too many to draw. Only its 199 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:
(3,7)(6,8)(9,10)(11,13)(12,15)(14,16)(17,18,20,22)(19,23,24,21)(26,28)(30,31);(2,5)(6,8)(10,11)(14,15)(17,18,20,22)(19,21,24,23)(25,26,29,28)(27,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,21)(20,22)(23,24)(25,27,29,32)(26,30,28,31);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,19,20,24)(18,21,22,23)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^8|a^2=efgh,b^2=ef,c^2=e,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| Fitting subgroup of | #15198⋊C3 |
| maximal subgroup of | #401975, #401976, #401979, … |
| maximal normal subgroup of | #401975, #401976, #401979, … |
| maximal quotient of | #154173, #154397, #154421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8186);