Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10218 |
| 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 | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
244 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 244 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:
(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);(1,2,4,6)(3,8,7,5)(9,11,15,24)(10,16,20,31)(12,26,28,39)(13,25,14,23)(17,33,35,37)(18,32,19,21)(22,29,38,27)(30,36,40,34);(1,2)(3,8)(4,6)(5,7)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,34)(19,36)(20,35)(21,33)(23,26)(24,38)(25,39)(31,40)(32,37)Polycyclic presentation:
2^8|a^2=gh,b^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fgh,d^c=dh,e^c=eh,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10226, #233829, #233832, … |
| maximal normal subgroup of | C2×#10226, #233829, #233832, … |
| maximal quotient of | #102549, #102562, #102565, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3214);