Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
215 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 215 conjugacy classes of subgroups — too many to draw. Only its 79 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,4)(5,6,7,8)(9,10)(11,22)(12,27)(13,30)(14,32)(15,33)(16,21)(17,24)(18,28)(19,23)(20,31)(25,35)(26,40)(29,37)(34,38)(36,39);(1,2,3,4)(6,8)(9,11,23,38,15,26,13,24,16,12,28,39,31,29,14,25)(10,17,32,40,20,34,18,22,21,35,30,37,33,36,19,27);(9,12,28,38,15,29,14,24,16,11,23,39,31,26,13,25)(10,17,30,37,20,34,19,27,21,35,32,40,33,36,18,22)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,c^2=egh,d^2=gh,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #93923, #93925, #152535, … |
| maximal normal subgroup of | #93923, #93925, #152535, … |
| maximal quotient of | #140727, #140771, #141206, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6228);