Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #353 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #127 | C23 |
| socle | C2 | C2×#353 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
208 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 208 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)(3,12,14,7)(4,8)(5,10,19,9)(6,11)(13,26,27,20)(15,21,18,22)(16,24,30,23)(17,25)(28,31,29,32);(1,3)(2,7)(4,13)(5,18)(6,15)(8,20)(9,21)(10,12)(11,22)(14,19)(16,29)(17,28)(23,31)(24,26)(25,32)(27,30);(1,4,6,17)(2,8,11,25)(3,13,15,28)(5,16,19,30)(7,20,22,32)(9,23,10,24)(12,26,21,31)(14,27,18,29)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=gh,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,d^b=dgh,e^b=efg,f^b=fh,e^d=egh,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94776, #94780, #94797, … |
| maximal normal subgroup of | #94776, #94780, #94797, … |
| maximal quotient of | #149573, #149623, #149690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6571);