Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8674 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #380 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
192 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 67 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:
(1,2,5,8,6,4,3,7)(9,10,14,17,15,12,13,23)(11,18,21,19,22,24,20,16);(1,3,6,5)(2,7,4,8)(9,11,13,20,15,22,14,21)(10,16,17,18,12,19,23,24);(1,4)(2,6)(3,8)(5,7)(9,12)(10,15)(11,19)(13,23)(14,17)(16,22)(18,21)(20,24)Polycyclic presentation:
2^8|a^2=f,b^2=dh,d^2=g,e^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,e^c=egh,f^c=fg46 conjugacy classes — view the full character table.
| maximal subgroup of | #131662, #131663, #131694, … |
| maximal normal subgroup of | #131662, #131663, #131694, … |
| maximal quotient of | #98892, #98893, #98898, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1684);