Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
276 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 123 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)(3,10,12,16)(4,8)(5,14)(6,9)(7,11,15,13)(17,18,21,25,22,26,20,24)(19,23,27,30,28,31,29,32);(1,3,5,13,6,12,4,11)(2,7,8,10,9,15,14,16)(17,19)(18,23)(20,29)(21,27)(22,28)(24,32)(25,30)(26,31);(17,20,22,21)(18,24,26,25)(19,27,28,29)(23,30,31,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,d^2=g,e^2=h,f^2=g,b^a=bd,f^a=fg,c^b=ch,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #105591, #105693, #105883, … |
| maximal normal subgroup of | #105591, #105693, #105883, … |
| maximal quotient of | #94865, #94866, #94869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 852);