Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#124 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#394 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
676 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 165 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)(3,12)(4,17)(5,22)(6,10)(7,11)(8,14)(9,20)(13,23)(15,25)(16,24)(18,26)(19,30)(21,27)(28,31)(29,32);(1,3)(2,8)(4,18)(5,15)(6,14)(7,16)(9,26)(10,24)(11,25)(12,22)(13,21)(17,23)(19,29)(20,28)(27,32)(30,31);(1,4,5,20,7,19,6,21)(2,9,10,17,11,27,22,30)(3,13,14,29,16,28,15,18)(8,23,12,26,25,31,24,32);(4,19)(9,27)(13,28)(17,30)(18,29)(20,21)(23,31)(26,32)Polycyclic presentation:
2^8|c^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=cg,e^b=eh,g^b=gh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #142306, #142307, #394630, … |
| maximal normal subgroup of | #142306, #142307, #394630, … |
| maximal quotient of | #238813, #238814, #238815, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16523);