Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #95 |
| commutator | C2×C4 | C4×C8 |
| Frattini, cosocle | C43 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
147 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(2,6)(3,4)(7,8)(9,10,11,13,12,14,15,16)(17,18)(19,25,27,31)(20,23)(21,24)(22,26,29,32)(28,30);(1,2,4,7,5,8,3,6)(17,19)(18,22,24,29)(20,26)(21,27)(23,25,30,31)(28,32)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=f,b^a=bc,c^a=cf,e^a=efg,d^b=dg64 conjugacy classes — view the full character table.
| maximal subgroup of | #92835, #92836, C2×#7018, … |
| maximal normal subgroup of | #92835, #92836, C2×#7018, … |
| maximal quotient of | #92806, #92809, #93304, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6);