Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3512 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
526 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 123 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 — 4 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,20)(12,24)(13,18)(14,28)(15,19)(16,22)(17,27)(21,38)(23,32)(25,31)(26,36)(29,33)(30,35)(34,39)(37,40);(2,5)(6,7)(9,11)(10,16)(12,25)(13,22)(14,23)(15,29)(17,34)(18,20)(19,32)(21,27)(24,37)(26,40)(28,33)(30,39)(31,36)(35,38);(9,12,14,27,13,26,15,30)(10,17,19,24,18,35,28,36)(11,21,23,40,22,39,29,25)(16,31,32,34,20,37,33,38);(1,3,4,8)(2,6,5,7)(10,18)(12,26)(16,20)(19,28)(21,39)(25,40)(27,30)(32,33)Polycyclic presentation:
2^8|c^2=f,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #268300, #268305, #268329, … |
| maximal normal subgroup of | #268300, #268305, #268329, … |
| maximal quotient of | #293533, #293538, #293541, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18607);