Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7447 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1361 |
| commutator | C2×C16 | C2×C4 |
| Frattini, cosocle | C4×C16 | C22 |
| prime radical | #2246 | C2 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
73 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,10,20,13,22,29,7,14,21,12,6,23,5,11)(3,15,17,9,28,27,32,26,19,30,31,25,18,8,16,24);(2,8)(3,16)(4,5)(6,20)(9,12)(10,25)(11,24)(13,30)(14,27)(15,23)(17,18)(19,32)(21,22)(26,29)(28,31)Polycyclic presentation:
2^8|a^2=d,c^2=fgh,d^2=efgh,e^2=gh,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fg,g^a=gh,c^b=cf,d^b=defg,e^b=eg,f^b=fgh,g^b=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7448, #150950, #150977, … |
| maximal normal subgroup of | C2×#7448, #150950, #150977, … |
| maximal quotient of | #93864, #93869, #93873, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 436);