Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19621 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3169 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
397 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 397 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,8)(3,4,7,5)(9,10)(11,17)(12,14)(13,21)(15,20)(16,18)(19,23)(22,24);(1,3,5,2,6,7,4,8)(10,14)(15,22)(16,21)(17,23);(9,11,18,24,12,19,13,20)(10,15,21,23,14,22,16,17);(1,4,6,5)(2,7,8,3)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=fh,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dg52 conjugacy classes — view the full character table.
| maximal subgroup of | #143442, #143444, #143454, … |
| maximal normal subgroup of | #143442, #143444, #143454, … |
| maximal quotient of | #191438, #206113, #206121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12627);