Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
391 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 391 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,19,21,25,23,26)(20,28,31,29,30,27,32,24);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,25,21,26,23,18)(20,29,31,27,30,24,32,28);(3,7)(6,8)(10,14)(13,16)(17,20)(18,24)(19,27)(21,30)(22,32)(23,31)(25,29)(26,28);(1,3,4,7)(2,6,5,8)(20,30)(24,29)(27,28)(31,32)Polycyclic presentation:
2^8|a^2=eg,b^2=f,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #170255, #170257, #170287, … |
| maximal normal subgroup of | #170255, #170257, #170287, … |
| maximal quotient of | #380163, #380607, #381038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23209);