Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10218 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
242 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10)(3,11,14,15)(4,9,5,8)(7,13,16,12)(19,26)(20,21)(23,27)(24,30)(28,29)(31,32);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,15,9,11)(17,18)(19,27)(20,24)(21,30)(22,25)(23,29)(26,31)(28,32);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,19)(18,23)(20,29)(21,26)(22,28)(24,32)(25,31)(27,30)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fgh,d^c=dh,e^c=eh,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10223, #233825, #233829, … |
| maximal normal subgroup of | C2×#10223, #233825, #233829, … |
| maximal quotient of | #102544, #102554, #102563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3211);