Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10144 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
264 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,7)(5,9)(6,10)(8,12)(13,15)(14,16)(19,25)(20,27)(22,30)(23,28)(26,29)(31,32);(2,7)(3,12)(4,5)(9,10)(11,15)(13,14)(17,18,21,24)(19,22,26,31)(20,28,27,23)(25,32,29,30);(1,3,6,13)(2,8,10,16)(4,12,5,14)(7,15,9,11)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,27)(28,30)Polycyclic presentation:
2^8|b^2=h,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,c^b=cf,d^b=dg,e^b=egh,f^b=fg,d^c=dh,e^c=egh,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10149, #236913, #236923, … |
| maximal normal subgroup of | C2×#10149, #236913, #236923, … |
| maximal quotient of | #102595, #102610, #102615, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3137);