Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10518 |
| automorphism | Aut(G) = C23×D43 (order 4096) · Inn(G) = #394 · Out(G) = C26 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
210 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,6,8)(3,5,7,4)(9,10,14,18)(11,12,16,22)(13,19,24,17)(15,23,21,20);(2,3)(4,5)(7,8)(9,11)(10,15)(12,19)(13,23)(14,16)(17,22)(18,21)(20,24);(1,3,5,8,6,7,4,2)(9,12,13,21,14,22,24,15)(10,16,17,23,18,11,19,20)Polycyclic presentation:
2^8|a^2=g,c^2=dh,d^2=g,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,e^c=egh,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10521, #238870, #238871, … |
| maximal normal subgroup of | C2×#10521, #238870, #238871, … |
| maximal quotient of | #104860, #104863, #104869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3509);