Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
592 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(1,2,4,6)(3,5,8,7)(14,20)(21,24);(2,5)(3,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,20)(19,22)(23,24);(9,11)(10,14)(12,17)(13,18)(15,21)(16,20)(19,23)(22,24);(2,6)(5,7)(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,21,20,24)Polycyclic presentation:
2^8|a^2=g,d^2=h,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #246928, #246929, #246936, … |
| maximal normal subgroup of | #246928, #246929, #246936, … |
| maximal quotient of | #165904, #165942, #165998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11141);