Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| 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 | #2218 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
258 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 103 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,22)(13,27)(14,29)(15,19)(17,26)(18,24)(20,34)(21,33)(23,36)(25,37)(28,40)(30,35)(31,38)(32,39);(2,5)(6,8)(9,11,15,14)(10,16,19,29)(12,23,25,20)(13,21,28,24)(17,31,32,30)(18,27,33,40)(22,36,37,34)(26,38,39,35);(9,12,24,38,14,23,28,39,15,25,21,35,11,20,13,26)(10,17,27,34,16,30,33,37,19,32,40,36,29,31,18,22);(1,3,4,7)(2,6,5,8)(12,25)(17,32)(20,23)(22,37)(26,39)(30,31)(34,36)(35,38)Polycyclic presentation:
2^8|b^2=h,c^2=fg,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,c^b=ch,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #150131, #150390, #150979, … |
| maximal normal subgroup of | #150131, #150390, #150979, … |
| maximal quotient of | #494201, #495307, #495308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26738);