Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7092 |
| automorphism | Aut(G) = #72279644 (order 2048) · Inn(G) = #331 · Out(G) = C2×C42 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #331 |
| commutator | C4×C8 | C2×C4 |
| Frattini, cosocle | C82 | C22 |
| prime radical | #493 | C4 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
79 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,8,11,6,10,14,5,9,13,16,4,7,12,15)(17,18)(19,22)(20,29)(21,25)(23,26)(24,27)(28,31)(30,32);(2,6)(3,10)(4,11)(7,9)(8,16)(12,13)(18,22)(19,26)(20,27)(23,24)(25,31)(28,32)(29,30)Polycyclic presentation:
2^8|a^2=d,c^2=fh,d^2=efg,e^2=g,f^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fg,c^b=cf,d^b=defg,e^b=eg,f^b=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7094, #122896, #122897, … |
| maximal normal subgroup of | C2×#7094, #122896, #122897, … |
| maximal quotient of | #92866, #92867, #93063, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 82);