Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#553 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
511 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 511 conjugacy classes of subgroups — too many to draw. Only its 179 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,26,22,25,20,24)(19,27,28,32,29,23,30,31);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,20,22,21)(18,24,25,26)(19,28,29,30)(23,31,27,32);(4,12)(7,14)(10,16)(13,15)(17,20,22,21)(18,24,25,26)(19,28,29,30)(23,31,27,32)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=h,f^2=h,b^a=be,e^a=eg,f^b=fg,d^c=dgh55 conjugacy classes — view the full character table.
| maximal subgroup of | #138792, #138794, #138808, … |
| maximal normal subgroup of | #138792, #138794, #138808, … |
| maximal quotient of | #164338, #164340, #164364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10412);