Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7422 |
| automorphism | Aut(G) = #72285396 (order 2048) · Inn(G) = #1347 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1347 |
| commutator | C2×C16 | C2×C4 |
| Frattini, cosocle | C2×M32 | C22 |
| prime radical | #2218 | C2 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ C2×M32 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
75 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)(3,13,16,7)(4,18)(5,21)(6,12)(8,14,26,29)(9,19)(10,15)(11,22)(17,25,32,28)(20,27)(23,30,24,31);(2,7)(3,14)(4,15)(5,22)(8,10)(9,26)(11,28)(12,13)(16,29)(17,31)(18,24)(19,20)(21,25)(23,27)(30,32)Polycyclic presentation:
2^8|a^2=d,c^2=fgh,e^2=g,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fgh,g^a=gh,c^b=cf,d^b=defgh,e^b=egh,f^b=fgh,g^b=gh,d^c=dh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7422, #150254, #150390, … |
| maximal normal subgroup of | C2×#7422, #150254, #150390, … |
| maximal quotient of | #93784, #93806, #93848, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 410);