Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18746 |
| 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 | C42 | C24 |
| prime radical | #3480 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
479 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 479 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(2,6)(3,4)(7,8)(9,10,15,20)(11,21,23,17)(12,24,22,16)(13,19,14,18);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,13)(14,22)(15,23)(17,18)(19,21)(20,24);(1,3,5,4)(2,6,7,8)(9,12,15,22)(10,17,20,21)(11,13,23,14)(16,18,24,19);(9,13,15,14)(10,18,20,19)(11,12,23,22)(16,17,24,21)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,e^b=eg,f^b=fh,e^c=eh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #270625, #270626, #270665, … |
| maximal normal subgroup of | #270625, #270626, #270665, … |
| maximal quotient of | #165954, #165959, #165966, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11790);