Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#127 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#399 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
494 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 494 conjugacy classes of subgroups — too many to draw. Only its 165 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,11,7,21)(3,12,14,24,16,28,15,8)(4,17,20,27,19,9,22,26)(13,29,18,32,30,25,31,23);(1,3,7,14,6,16,5,15)(2,8,21,12,11,24,10,28)(4,18,22,30,19,31,20,13)(9,25,27,23,17,29,26,32);(1,4)(2,9)(3,13)(5,20)(6,19)(7,22)(8,23)(10,26)(11,17)(12,29)(14,18)(15,31)(16,30)(21,27)(24,32)(25,28);(4,19)(9,17)(13,30)(18,31)(20,22)(23,32)(25,29)(26,27)Polycyclic presentation:
2^8|a^2=e,b^2=ef,e^2=h,g^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #142398, #142399, #396532, … |
| maximal normal subgroup of | #142398, #142399, #396532, … |
| maximal quotient of | #243854, #243855, #243856, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16728);