Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9183 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
306 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 67 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 — 3 generators on 32 points:
(1,2,5,11,20,27,4,19)(3,13,17,25,31,32,15,26)(6,10,22,28,7,12,21,9)(8,18,24,23,29,16,30,14);(1,3)(2,8,10,25)(4,14)(5,23)(6,16)(7,18)(9,24,19,32)(11,13,28,30)(12,26,27,29)(15,22)(17,21)(20,31);(1,4,7,22)(2,9,12,11)(3,14,18,15)(5,6,21,20)(8,24,26,13)(10,19,27,28)(16,17,31,23)(25,32,29,30)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=h,d^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fh,d^c=dg,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | #133005, #133006, #133007, … |
| maximal normal subgroup of | #133005, #133006, #133007, … |
| maximal quotient of | #99821, #99825, #99835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2189);