Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C23 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
247 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 247 conjugacy classes of subgroups — too many to draw. Only its 111 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,20,25)(19,27,22,23)(21,24,29,32)(26,28,31,30);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,20,22)(18,23,25,27)(21,28,29,30)(24,26,32,31);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,24)(19,22)(21,30)(23,31)(25,32)(26,27)(28,29);(2,8)(6,9)(7,16)(14,15)(18,25)(23,27)(24,32)(26,31)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=eh,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,c^b=ce,f^c=fg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #139012, #139017, #139073, … |
| maximal normal subgroup of | #139012, #139017, #139073, … |
| maximal quotient of | #496428, #496477, #496733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26784);