Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | Aut(G) order 8192 |
| 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 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
267 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 267 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)(9,10,15,20)(11,16,23,34)(12,18,14,30)(13,26,29,40)(17,35,37,22)(19,25,38,28)(21,31,24,33)(27,32,39,36);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,24)(13,22)(14,21)(15,23)(17,26)(18,33)(19,32)(20,34)(25,39)(27,28)(29,35)(30,31)(36,38)(37,40);(9,11)(10,17)(12,21)(13,27)(14,24)(15,23)(16,26)(18,32)(19,33)(20,37)(22,28)(25,35)(29,39)(30,36)(31,38)(34,40);(9,12,15,14)(10,18,20,30)(11,21,23,24)(13,25,29,28)(16,31,34,33)(17,32,37,36)(19,26,38,40)(22,27,35,39)Polycyclic presentation:
2^8|a^2=eh,b^2=e,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #503647, #503649, #503651, … |
| maximal normal subgroup of | #503647, #503649, #503651, … |
| maximal quotient of | #493766, #494288, #494292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26622);