Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| 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 | #121 |
| 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 | |
253 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 253 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:
(2,4)(5,6,7,10,8,11,12,9)(13,14)(15,23)(16,19)(17,26)(18,22)(20,24)(21,27)(25,28);(1,2,3,4)(6,9)(7,12)(10,11)(13,15,17,16,18,24,27,25)(14,19,21,23,22,28,26,20);(13,16,17,24,18,25,27,15)(14,20,21,19,22,23,26,28)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=e,d^2=gh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cg,d^b=dh,f^b=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136900, #136901, #136908, … |
| maximal normal subgroup of | #136900, #136901, #136908, … |
| maximal quotient of | #102004, #102028, #102239, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2730);