Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| 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 | #3341 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
240 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 240 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:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,22)(13,25)(14,29)(15,19)(17,23)(18,27)(20,36)(21,35)(24,38)(26,31)(28,39)(30,34)(32,40)(33,37);(1,3)(2,6)(4,7)(5,8)(9,11,15,14)(10,16,19,29)(12,20,24,30)(13,21,28,27)(17,31,33,32)(18,25,35,39)(22,36,38,34)(23,26,37,40);(1,4,5,2)(3,7,8,6)(9,12)(10,17)(11,20)(13,26)(14,30)(15,24)(16,31)(18,34)(19,33)(21,37)(22,25)(23,27)(28,40)(29,32)(35,36)(38,39);(3,8)(6,7)(9,11,15,14)(10,16,19,29)(12,20,24,30)(13,21,28,27)(17,31,33,32)(18,25,35,39)(22,36,38,34)(23,26,37,40)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^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=de,f^c=fg,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #491909, #491910, #491916, … |
| maximal normal subgroup of | #491909, #491910, #491916, … |
| maximal quotient of | #497542, #497543, #497773, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26895);