Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #120 |
| 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 | |
275 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 275 conjugacy classes of subgroups — too many to draw. Only its 87 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)(7,9)(8,11)(10,12)(13,14,17,19,18,22,16,21)(15,23,25,20,26,27,24,28);(1,2,3,4)(14,19)(16,17)(20,23)(21,22)(24,25)(27,28);(6,9)(7,12)(10,11)(13,15,18,26)(14,20,22,28)(16,25,17,24)(19,23,21,27)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egh,f^c=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9058, #195546, #195555, … |
| maximal normal subgroup of | C2×#9058, #195546, #195555, … |
| maximal quotient of | #99596, #99606, #99632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2046);