Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| 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 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | C23 | #102 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
179 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 179 conjugacy classes of subgroups — too many to draw. Only its 65 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:
(1,2,3,6,4,5,7,8)(9,10)(11,20)(12,17)(13,26)(14,30)(15,19)(16,22)(18,28)(21,34)(23,40)(24,37)(25,35)(27,33)(29,36)(31,38)(32,39);(9,11,15,24)(10,16,19,32)(12,23,14,21)(13,27,29,38)(17,33,30,31)(18,34,35,40)(20,36,37,26)(22,25,39,28);(2,5)(6,8)(9,12,15,14)(10,17,19,30)(11,21,24,23)(13,25,29,28)(16,31,32,33)(18,26,35,36)(20,34,37,40)(22,38,39,27)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,d^b=df,f^b=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #131037, #131051, #131060, … |
| maximal normal subgroup of | #131037, #131051, #131060, … |
| maximal quotient of | #124667, #124704, #124737, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4650);