Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| 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 | C23 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
237 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 237 conjugacy classes of subgroups — too many to draw. Only its 135 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)(11,19)(12,20)(13,23)(15,25)(17,26)(18,30)(21,35)(22,33)(24,34)(27,36)(28,40)(29,39)(31,32)(37,38);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,25)(20,26)(21,28)(22,27)(23,30)(24,29)(31,37)(32,38)(33,36)(34,39)(35,40);(9,11,20,32,13,21,24,33,14,22,34,35,23,31,12,19)(10,15,26,38,18,28,29,36,16,27,39,40,30,37,17,25);(10,16)(15,27)(17,29)(18,30)(25,36)(26,39)(28,37)(38,40)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=fg,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #153216, #153217, #153325, … |
| maximal normal subgroup of | #153216, #153217, #153325, … |
| maximal quotient of | #494165, #495025, #495027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26717);