Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9183 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
265 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 265 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)(11,14,18,22)(12,19,20,15)(13,16)(17,21)(23,24);(1,2,3,7,4,6,8,5)(9,11)(10,14,16,22)(12,17)(13,18)(15,21,19,24)(20,23);(9,12,13,20)(10,15,16,19)(11,17,18,23)(14,21,22,24)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,d^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fh,d^c=dg,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | #131129, #131130, #131131, … |
| maximal normal subgroup of | #131129, #131130, #131131, … |
| maximal quotient of | #99820, #99821, #99828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2172);