Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10128 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
268 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,15,20)(11,21,24,32)(12,17,28,35)(13,19,14,18)(16,25,31,23)(22,37,38,33)(26,40,39,30)(27,36,29,34);(3,7)(5,8)(9,11)(10,16)(12,26)(13,25)(14,23)(15,24)(17,33)(18,32)(19,21)(20,31)(22,27)(28,39)(29,38)(30,34)(35,37)(36,40);(1,2)(3,8)(4,6)(5,7)(9,12,15,28)(10,17,20,35)(11,22,24,38)(13,27,14,29)(16,30,31,40)(18,34,19,36)(21,33,32,37)(23,26,25,39)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,c^b=cf,d^b=dg,e^b=egh,f^b=fg,d^c=dh,e^c=eh,f^c=fg31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10133, #236711, #236728, … |
| maximal normal subgroup of | C2×#10133, #236711, #236728, … |
| maximal quotient of | #102651, #102654, #102659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3121);