Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | Aut(G) order 4096 |
| 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 | C2×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
198 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,10,15)(7,11,19,28)(8,12,23,31)(9,24,22,14)(13,32,30,21)(16,26,18,27)(17,35,20,29)(25,36,33,34);(1,2)(3,4)(5,7)(6,11)(8,20)(9,16)(10,19)(12,29)(13,33)(14,27)(15,28)(17,23)(18,22)(21,36)(24,26)(25,30)(31,35)(32,34);(1,3,4,2)(6,12)(7,16)(8,21)(9,22)(11,25)(13,14)(15,31)(17,20)(18,19)(23,32)(24,30)(26,35)(27,29)(28,33)Polycyclic presentation:
2^8|a^2=h,c^2=d,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,e^b=eh,f^b=fh,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13388, #499019, #499021, … |
| maximal normal subgroup of | C2×#13388, #499019, #499021, … |
| maximal quotient of | #144684, #144685, #144788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6376);