Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| 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 | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
244 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,6,8,5,3,4,7)(9,10)(11,20)(12,24,27,34)(13,28)(14,18)(15,19)(16,22)(17,26,33,25)(21,36,38,30)(23,31)(29,32)(35,37,40,39);(1,3,5,2)(4,7,6,8)(9,11,13,22,15,23,14,29)(10,16,18,20,19,32,28,31)(12,21,26,39,27,38,25,37)(17,30,34,35,33,36,24,40);(4,6)(7,8)(9,12)(10,17)(11,21)(13,26)(14,25)(15,27)(16,30)(18,34)(19,33)(20,35)(22,39)(23,38)(24,28)(29,37)(31,40)(32,36)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,d^2=gh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eg,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94790, #94794, #152193, … |
| maximal normal subgroup of | #94790, #94794, #152193, … |
| maximal quotient of | #100095, #100096, #100105, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2409);