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 | C23 | #127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
375 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 375 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(2,5)(3,7)(6,8)(9,10,13,16)(11,17,20,24)(12,21,18,15)(14,22,23,19);(1,2,3,5,4,6,7,8);(9,11,18,22,13,20,12,19)(10,14,21,24,16,23,15,17);(2,6)(5,8)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=fh,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 | #143443, #143445, #143455, … |
| maximal normal subgroup of | #143443, #143445, #143455, … |
| maximal quotient of | #191436, #206118, #206121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12620);