Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | Aut(G) order 32768 |
| 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 |
| prime radical | C22×D8 | C22 |
| 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 119 normal subgroups are shown.
This group has 299 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) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,23)(12,17)(13,31)(14,34)(15,35)(16,22)(18,25)(19,24)(20,33)(21,32)(26,38)(27,37)(28,36)(29,40)(30,39);(2,5)(6,8)(9,11,24,30,15,27,33,28,16,25,14,26,32,29,13,12)(10,17,31,40,21,38,34,18,22,36,20,37,35,39,19,23);(1,3,4,7)(2,6,5,8)(9,12,13,29,32,26,14,25,16,28,33,27,15,30,24,11)(10,18,19,38,35,40,20,17,22,23,34,39,21,37,31,36);(11,25)(12,28)(17,36)(18,23)(26,30)(27,29)(37,40)(38,39)Polycyclic presentation:
2^8|b^2=fh,c^2=e,e^2=gh,f^2=g,g^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fgh,g^a=gh,d^b=dh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137515, #143336, #143341, … |
| maximal normal subgroup of | #137515, #143336, #143341, … |
| maximal quotient of | #494088, #494915, #494916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26694);