Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18866 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C23 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
439 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 439 conjugacy classes of subgroups — too many to draw. Only its 129 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 — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10,15,20)(11,21,23,32)(12,26,29,33)(13,19,14,18)(16,25,31,24)(17,27,34,28)(22,36,37,30)(35,38,40,39);(1,2,5,8)(3,6,4,7)(9,11,15,23)(10,16,20,31)(12,22,29,37)(13,25,14,24)(17,30,34,36)(18,32,19,21)(26,40,33,35)(27,39,28,38);(9,12,15,29)(10,17,20,34)(11,22,23,37)(13,28,14,27)(16,30,31,36)(18,26,19,33)(21,35,32,40)(24,39,25,38);(1,3,5,4)(2,7,8,6)(11,23)(16,31)(21,32)(22,37)(24,25)(30,36)(35,40)(38,39)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=h,d^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #250592, #250594, #250608, … |
| maximal normal subgroup of | #250592, #250594, #250608, … |
| maximal quotient of | #167254, #167258, #167263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11912);