Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C22×D8 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
359 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 157 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 — 4 generators on 24 points:
(1,2,4,7,5,3,6,8)(9,10,13,16)(11,18,19,15)(12,14,21,17)(20,23,24,22);(10,14)(12,21)(15,22)(16,17)(18,23)(20,24);(1,3,4,8,5,2,6,7)(9,11)(10,15)(12,20)(13,19)(14,22)(16,18)(17,23)(21,24);(2,3)(7,8)(10,16)(11,19)(14,17)(20,24)Polycyclic presentation:
2^8|a^2=f,c^2=fg,e^2=g,f^2=h,b^a=be,c^a=cg,d^a=dgh,e^a=eg,e^b=eg,d^c=dgh58 conjugacy classes — view the full character table.
| maximal subgroup of | #136478, #137021, #137023, … |
| maximal normal subgroup of | #136478, #137021, #137023, … |
| maximal quotient of | #183193, #183661, #183961, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11052);