Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18162 |
| automorphism | Aut(G) order 16384 |
| 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 | #2397 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
553 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 553 conjugacy classes of subgroups — too many to draw. Only its 117 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24);(1,2,4,6)(3,8,7,5)(9,11,14,21)(10,15,17,23)(12,20,13,19)(16,24,22,18);(3,7)(5,8);(2,6)(5,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)Polycyclic presentation:
2^8|b^2=g,d^2=gh,e^2=g,b^a=be,c^a=cf,e^a=eg,d^b=dg,e^b=eg,f^b=fh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139962, #139964, #139970, … |
| maximal normal subgroup of | #139962, #139964, #139970, … |
| maximal quotient of | #166327, #166383, #166439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11168);