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 | |
100 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(17,18,21,24)(19,26,27,31)(20,29,30,32)(22,28,23,25);(2,6)(7,15)(8,13)(9,16)(17,19,26,24,22,29,20,28,21,27,31,18,23,32,30,25);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9)(18,24)(19,27)(25,28)(29,32)Polycyclic presentation:
2^8|a^2=h,b^2=dfgh,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×#12482, #490629, #490631, … |
| maximal normal subgroup of | C2×#12482, #490629, #490631, … |
| maximal quotient of | #135012, #138593, #138672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5470);