Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
185 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 185 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(1,2,4,6)(3,7,8,5)(9,10)(11,19)(12,23)(13,28)(14,17)(15,20)(16,27)(18,26)(21,39)(22,36)(24,35)(25,34)(29,33)(30,37)(31,40)(32,38);(1,3,4,8)(2,5,6,7)(12,24)(15,30)(16,32)(20,37)(23,35)(25,40)(27,38)(31,34);(9,11,14,22)(10,15,17,30)(12,25,24,40)(13,29,26,21)(16,33,32,39)(18,34,28,31)(19,35,36,23)(20,38,37,27)Polycyclic presentation:
2^8|a^2=d,b^2=d,c^2=h,e^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,e^b=eh,e^c=eg,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #125885, #125886, #125928, … |
| maximal normal subgroup of | #125885, #125886, #125928, … |
| maximal quotient of | #144445, #144447, #144476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6356);