Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C2×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1149 conjugacy classes of subgroups; only the 479 normal subgroups are shown.
This group has 1149 conjugacy classes of subgroups — too many to draw. Only its 479 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) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(11,19)(12,13)(15,22)(17,18)(20,21)(23,24);(2,5)(6,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,23)(21,24);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,22)(18,24);(10,16)(15,23)(17,18)(22,24);(9,12,14,13)(10,17,16,18)(11,19,20,21)(15,22,23,24)Polycyclic presentation:
2^8|c^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,d^b=dh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #147614, #147616, #246327, … |
| maximal normal subgroup of | #147614, #147616, #246327, … |
| maximal quotient of | #6370887, #6371276, #6371387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53506);