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) = C24×C4 |
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×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
504 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 504 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)(17,18,21,25,22,26,20,24)(19,23,27,30,28,31,29,32);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,22,28)(18,23,26,31)(20,29,21,27)(24,32,25,30);(17,20,22,21)(18,24,26,25)(19,27,28,29)(23,30,31,32)Polycyclic presentation:
2^8|a^2=f,c^2=gh,d^2=h,f^2=h,b^a=be,e^a=eg,f^b=fg,d^c=dh55 conjugacy classes — view the full character table.
| maximal subgroup of | #138797, #138799, #138813, … |
| maximal normal subgroup of | #138797, #138799, #138813, … |
| maximal quotient of | #164138, #164140, #164141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10403);