Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12433 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #466 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
97 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(18,22)(19,26)(20,27)(23,28)(24,29)(25,32)(30,31);(17,18,23,32,20,25,19,22,21,24,26,31,27,30,28,29);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(18,24)(22,29)(25,30)(31,32)Polycyclic presentation:
2^8|a^2=g,b^2=dfh,c^2=g,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,c^b=ch,e^c=eg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12462, #490610, #490611, … |
| maximal normal subgroup of | C2×#12462, #490610, #490611, … |
| maximal quotient of | #134978, #138593, #138594, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5450);