Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1267 conjugacy classes of subgroups; only the 411 normal subgroups are shown.
This group has 1267 conjugacy classes of subgroups — too many to draw. Only its 411 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) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,4,8,6)(9,10,12,13)(11,14,15,16)(17,18)(19,21)(20,22)(23,24);(1,3,5,8)(2,6,7,4)(10,13)(11,15)(17,18)(19,23)(20,22)(21,24);(9,11,12,15)(10,14,13,16)(17,18,20,22)(19,21,24,23);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24);(1,4)(2,3)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,24)(22,23)Polycyclic presentation:
2^8|a^2=f,b^2=fgh,c^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #47563⋊C3, #72283819, #72283832 |
| maximal subgroup of | #130056, #133431, #251855, … |
| maximal normal subgroup of | #130056, #133431, #251855, … |
| maximal quotient of | #576597, #577210, #577897, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40551);