Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #127 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
239 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 239 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,14)(15,21)(16,24)(17,19)(18,22)(20,25)(23,27)(26,28);(2,4)(5,6,10,12,9,11,8,7)(13,15,16,22,17,23,25,28)(14,18,20,21,19,26,24,27);(5,7,10,6,9,12,8,11)(14,19)(18,26)(20,24)(21,27)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=eg,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9840, #206165, #206168, … |
| maximal normal subgroup of | C2×#9840, #206165, #206168, … |
| maximal quotient of | #102120, #102144, #102168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2828);