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) = C23 |
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 | |
153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(9,10,11,13,12,14,15,16)(17,18)(19,25,26,22)(20,24)(21,23)(27,31,32,29)(28,30);(2,5,7,6)(17,19,20,27,21,26,28,32)(18,22,24,29,23,25,30,31)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=g,b^a=bc,c^a=cf,e^a=efg,d^b=dg64 conjugacy classes — view the full character table.
| maximal subgroup of | #92855, #92856, C2×#7019, … |
| maximal normal subgroup of | #92855, #92856, C2×#7019, … |
| maximal quotient of | #92806, #92811, #93315, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7);