Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| 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 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
317 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 317 conjugacy classes of subgroups — too many to draw. Only its 119 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)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,25)(19,23,27,30)(20,29,24,21)(26,32,31,28);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19,22,27)(18,23,25,30)(20,26,24,31)(21,28,29,32);(1,3,5,4)(2,7,6,8)(17,20,21,18,22,24,29,25)(19,26,28,23,27,31,32,30);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(19,27)(23,30)(26,31)(28,32)Polycyclic presentation:
2^8|a^2=h,b^2=gh,c^2=ef,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #125093, #125270, #125410, … |
| maximal normal subgroup of | #125093, #125270, #125410, … |
| maximal quotient of | #392543, #393183, #393187, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26273);