Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13584 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#353 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C4⋊Q8 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
199 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 45 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:
(1,2,5,10,6,11,21,28)(3,12,14,27,15,22,30,31)(4,8,17,25,18,9,20,26)(7,19,23,16,24,29,32,13);(1,3,6,15)(2,7,11,24)(4,16,18,13)(5,19,21,29)(8,12,9,22)(10,27,28,31)(14,20,30,17)(23,25,32,26);(1,4,6,18)(2,8,11,9)(3,13,15,16)(5,17,21,20)(7,22,24,12)(10,25,28,26)(14,19,30,29)(23,31,32,27)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=gh,e^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=ch,d^b=dgh,e^b=efg,f^b=fh,e^d=egh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94789, #152052, #152090, … |
| maximal normal subgroup of | #94789, #152052, #152090, … |
| maximal quotient of | #149576, #149625, #149695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6580);