Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10603 |
| automorphism | Aut(G) = C23×D43 (order 4096) · Inn(G) = #396 · Out(G) = C26 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #396 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C22⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
153 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 153 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,8,5,4,6,7)(9,10)(11,19)(12,17)(13,16)(14,18)(15,21)(20,23)(22,24);(1,3,5,6)(2,7,4,8)(9,11,12,21,14,20,13,22)(10,15,16,19,18,24,17,23);(1,4,5,2)(3,8,6,7)(10,16)(11,20)(12,13)(15,19)(17,18)(23,24)Polycyclic presentation:
2^8|a^2=de,b^2=d,c^2=gh,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,f^b=fg,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10606, #242673, #242678, … |
| maximal normal subgroup of | C2×#10606, #242673, #242678, … |
| maximal quotient of | #105188, #105191, #105194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3594);