Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
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 | |
330 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 121 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)(3,6)(4,5)(7,8)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29);(9,10)(11,13)(12,15)(14,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31);(2,5)(6,8)(10,13)(11,16)(14,15)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31);(1,3,4,7)(2,6,5,8)(20,28)(23,30)(26,32)(29,31)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=e,d^2=f,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,d^c=de,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #100660, #103019, #103021, … |
| maximal normal subgroup of | #100660, #103019, #103021, … |
| maximal quotient of | #380492, #381201, #381404, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23079);