Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12436 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1667 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7)(3,12)(4,8)(5,17)(10,28)(11,22)(13,14)(15,29)(16,30)(18,23)(19,24)(20,21)(25,26)(31,32);(1,2,8,21,5,10,18,22,6,11,23,28,17,20,4,7)(3,9,12,24,14,25,29,32,16,27,30,31,15,26,13,19);(1,3)(2,9)(4,13)(5,14)(6,16)(7,19)(8,12)(10,25)(11,27)(15,17)(18,29)(20,26)(21,24)(22,32)(23,30)(28,31)Polycyclic presentation:
2^8|b^2=df,d^2=fh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gh,e^c=eg,g^c=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | #151524, #151525, #151528, … |
| maximal normal subgroup of | #151524, #151525, #151528, … |
| maximal quotient of | #123363, #123364, #123371, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5424);