Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | Aut(G) = #72338260 (order 4096) · Inn(G) = D42 · Out(G) = #392 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | D4×C4⋊C4 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
404 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 197 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,13,8,14,7,12)(3,15,19,27,20,31,18,9)(4,21,22,29,23,26,17,10)(5,11,25,30,16,28,24,32);(1,3)(2,9)(4,5)(6,19)(7,18)(8,20)(10,28)(11,29)(12,31)(13,15)(14,27)(16,23)(17,24)(21,32)(22,25)(26,30);(1,4,7,17,8,23,6,22)(2,10,13,21,14,29,12,26)(3,16,18,25,20,5,19,24)(9,11,15,30,27,28,31,32);(1,5)(2,11)(3,17)(4,19)(6,25)(7,24)(8,16)(9,26)(10,15)(12,32)(13,30)(14,28)(18,23)(20,22)(21,27)(29,31)Polycyclic presentation:
2^8|a^2=g,c^2=f,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ch,d^b=de,e^b=eh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #501814, #501815, #503870, … |
| maximal normal subgroup of | #501814, #501815, #503870, … |
| maximal quotient of | #225498, #225503, #225508, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15683);