Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #381 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | C22⋊#95 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
582 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 582 conjugacy classes of subgroups — too many to draw. Only its 173 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)(3,7,12,20)(4,14,15,8)(5,17)(6,9)(10,18)(11,23)(13,27,26,21)(16,22,29,28)(19,24)(25,30)(31,32);(1,3,5,13,6,12,18,26)(2,7,10,21,9,20,17,27)(4,11,16,25,15,24,29,32)(8,19,22,30,14,23,28,31);(1,4)(2,8)(3,11)(5,16)(6,15)(7,19)(9,14)(10,22)(12,24)(13,25)(17,28)(18,29)(20,23)(21,30)(26,32)(27,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=dh,f^b=fh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #146677, #146691, #146694, … |
| maximal normal subgroup of | #146677, #146691, #146694, … |
| maximal quotient of | #105608, #105609, #105610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3759);