Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19883 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
330 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 103 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)(3,7,5,4)(9,10,14,17)(11,15,13,21)(12,19,20,23)(16,22,24,18);(2,4)(3,5)(7,8)(9,11,14,13)(10,15,17,21)(12,18,20,22)(16,23,24,19);(1,3,6,5)(2,7,8,4)(9,12,14,20)(10,16,17,24)(11,18,13,22)(15,23,21,19);(1,4)(2,3)(5,8)(6,7)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=cg,d^b=de,e^b=eg,f^c=fh,e^d=eg43 conjugacy classes — view the full character table.
| maximal subgroup of | #491713, #491715, #491729, … |
| maximal normal subgroup of | #491713, #491715, #491729, … |
| maximal quotient of | #220786, #220957, #220976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12925);