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 | C2×C42 | C23 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
239 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 239 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)(9,10,13,17,14,16,23,11)(12,19,21,24,20,15,22,18);(1,2,3,7,4,6,8,5)(9,11,13,16,14,17,23,10)(12,18,22,19,20,24,21,15);(9,12,14,20)(10,15,16,19)(11,18,17,24)(13,22,23,21)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,d^2=gh,f^2=g,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 | #131134, #131136, #131137, … |
| maximal normal subgroup of | #131134, #131136, #131137, … |
| maximal quotient of | #99820, #99822, #99832, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2176);