Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
299 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 299 conjugacy classes of subgroups — too many to draw. Only its 101 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:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,21)(12,20)(13,16)(14,27)(15,18)(17,24)(19,29)(22,37)(23,39)(25,32)(26,34)(28,36)(30,35)(31,38)(33,40);(1,3,4,8)(2,5,6,7)(10,16)(11,22)(18,20)(19,32)(23,30)(24,38)(27,34)(28,40);(3,8)(5,7)(9,11)(10,17)(12,23)(13,22)(14,28)(15,30)(16,31)(18,33)(19,35)(20,36)(21,27)(24,29)(25,38)(26,40)(32,39)(34,37);(2,6)(5,7)(9,12,13,15)(10,18,16,20)(11,23,22,30)(14,25,26,29)(17,33,31,36)(19,34,32,27)(21,35,37,39)(24,28,38,40)Polycyclic presentation:
2^8|b^2=eh,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,c^b=ce,f^c=fg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #489355, #489356, #489383, … |
| maximal normal subgroup of | #489355, #489356, #489383, … |
| maximal quotient of | #496354, #496479, #496737, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26766);