Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | C23 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
153 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 153 conjugacy classes of subgroups — too many to draw. Only its 73 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 — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,14,20,15,21,30,37,16,22,28,35,12,18,13,19)(11,23,26,34,24,36,25,38,27,39,29,17,31,33,40,32);(1,2,4,6)(3,8,7,5)(9,11)(10,17)(12,24)(13,29)(14,25)(15,31)(16,27)(18,32)(19,36)(20,23)(21,38)(22,34)(26,30)(28,40)(33,37)(35,39);(9,12,16,15)(10,18,22,21)(11,24,27,31)(13,28,30,14)(17,32,34,38)(19,35,37,20)(23,36,39,33)(25,29,40,26)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,d^2=gh,e^2=fg,g^2=h,b^a=bd,d^a=df,d^b=dg,e^b=efg,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #128914, #128915, #128954, … |
| maximal normal subgroup of | #128914, #128915, #128954, … |
| maximal quotient of | #124083, #124231, #124232, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4922);