Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
271 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 271 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,21,23,38)(12,20,15,18)(14,27,26,35)(17,33,31,24)(19,25,32,29)(22,39,30,36)(28,37,40,34);(1,3,4,8)(2,5,6,7)(10,16)(17,31)(18,20)(19,32)(21,38)(27,35)(34,37)(36,39);(9,11)(10,17)(12,22)(13,23)(14,28)(15,30)(16,31)(18,34)(19,36)(20,37)(21,27)(24,29)(25,33)(26,40)(32,39)(35,38);(9,12,13,15)(10,18,16,20)(11,22,23,30)(14,25,26,29)(17,34,31,37)(19,35,32,27)(21,36,38,39)(24,28,33,40)Polycyclic presentation:
2^8|a^2=e,b^2=eh,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fg,g^a=gh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #138844, #138845, #138850, … |
| maximal normal subgroup of | #138844, #138845, #138850, … |
| maximal quotient of | #493763, #494285, #494788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26618);