Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9159 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = #386 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3115 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
236 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(2,5,6,7)(9,10)(11,14)(12,20,23,18)(13,19)(15,17)(16,22,24,21);(1,2,3,7,4,6,8,5)(9,11,14,10,15,19,13,17)(12,18,21,16,23,20,22,24);(9,12,15,23)(10,16,17,24)(11,18,19,20)(13,22,14,21)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=g,d^2=gh,e^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=eg,f^c=fg43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9162, #196988, #196990, … |
| maximal normal subgroup of | C2×#9162, #196988, #196990, … |
| maximal quotient of | #99792, #99793, #99800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2150);