Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9471 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
259 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 259 conjugacy classes of subgroups — too many to draw. Only its 61 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 — 3 generators on 32 points:
(1,2,5,9,6,10,4,8)(3,11,12,15,14,16,13,7)(17,18)(19,22)(20,27,28,23)(21,24)(25,29,31,32)(26,30);(1,3)(2,7,10,15)(4,13)(5,12)(6,14)(8,16,9,11)(17,19,21,26)(18,22,24,30)(20,25,28,31)(23,29,27,32);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,11,15,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,31)(30,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=g,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,e^b=eh,d^c=dh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132903, #132904, #132907, … |
| maximal normal subgroup of | #132903, #132904, #132907, … |
| maximal quotient of | #100168, #100170, #100176, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2470);