Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #121 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1797 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
220 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,13,7,11,18,26,8,14,17,9,5,12,3,15)(4,19,22,31,21,28,16,30,23,29,24,32,20,10,27,25);(1,3,5,17,8,18,7,6)(2,9,11,15,14,13,12,26)(4,16,20,22,23,27,21,24)(10,25,19,31,28,30,29,32);(1,4)(2,10)(3,16)(5,20)(6,24)(7,21)(8,23)(9,25)(11,19)(12,29)(13,30)(14,28)(15,31)(17,22)(18,27)(26,32)Polycyclic presentation:
2^8|a^2=f,b^2=d,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^c=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129525, #129526, #129998, … |
| maximal normal subgroup of | #129525, #129526, #129998, … |
| maximal quotient of | #134224, #134225, #134252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5780);