Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | Aut(G) = C22×#483875 (order 2048) · Inn(G) = #357 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #357 |
| commutator | C16 | C2×C8 |
| Frattini, cosocle | C4×C16 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,5,4,6,7,8)(10,15,16,30)(11,19)(12,24)(13,26)(17,34,33,39)(18,35,28,32)(20,25,31,29)(21,27)(22,40)(23,38)(36,37);(9,10,14,16)(11,20,23,31)(12,25,27,29)(13,28,26,18)(15,19,30,38)(17,21,33,24)(22,34,36,39)(32,40,35,37)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=egh,d^2=f,e^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=ceh,d^b=dh,e^b=eg,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7386, #150062, #150063, … |
| maximal normal subgroup of | C2×#7386, #150062, #150063, … |
| maximal quotient of | #93750, #93765, #93916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 374);