Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #381 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#381 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
622 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 622 conjugacy classes of subgroups — too many to draw. Only its 221 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 — 4 generators on 32 points:
(1,2,6,12,7,11,5,10)(3,13,15,26,16,8,21,23)(4,17,20,24,19,9,18,25)(14,22,28,31,29,27,30,32);(1,3)(2,8)(4,14)(5,15)(6,21)(7,16)(9,22)(10,23)(11,13)(12,26)(17,27)(18,28)(19,29)(20,30)(24,31)(25,32);(1,4)(2,9,11,17)(3,14)(5,18)(6,20)(7,19)(8,22,13,27)(10,24,12,25)(15,28)(16,29)(21,30)(23,31,26,32);(1,5,7,6)(2,10,11,12)(3,15,16,21)(4,18,19,20)(8,23,13,26)(9,24,17,25)(14,28,29,30)(22,31,27,32)Polycyclic presentation:
2^8|a^2=g,c^2=e,d^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,g^b=gh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #146499, #146502, #147612, … |
| maximal normal subgroup of | #146499, #146502, #147612, … |
| maximal quotient of | #190012, #190013, #190018, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13701);