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 | #3176 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
272 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 101 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)(10,15)(11,19)(13,24)(14,23)(16,30)(17,33)(20,38)(21,39)(22,37)(25,26)(27,31)(28,35)(32,34)(36,40);(2,5)(6,8)(9,10)(11,16)(12,15)(13,17)(14,18)(19,31)(20,27)(21,32)(22,28)(23,29)(24,35)(25,34)(26,36)(30,38)(33,37)(39,40);(9,11,12,20)(10,16,15,27)(13,25,22,39)(14,26,23,21)(17,34,28,40)(18,36,29,32)(19,37,38,24)(30,35,31,33);(1,3,4,7)(2,6,5,8)(10,15)(16,27)(17,28)(18,29)(30,31)(32,36)(33,35)(34,40)Polycyclic presentation:
2^8|c^2=h,d^2=eh,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 | C2×#33686, #7623095, #7623104, … |
| maximal normal subgroup of | C2×#33686, #7623095, #7623104, … |
| maximal quotient of | #494036, #494974, #494977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26674);