Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#124 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#400 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
324 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 149 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:
(1,2,5,10,6,11,7,22)(3,12,14,24,16,27,15,8)(4,17,20,31,19,9,21,26)(13,25,29,32,28,30,18,23);(1,3,7,14,6,16,5,15)(2,8,22,12,11,24,10,27)(4,18,21,13,19,29,20,28)(9,25,31,32,17,30,26,23);(1,4,5,20,6,19,7,21)(2,9,10,26,11,17,22,31)(3,13,14,29,16,28,15,18)(8,23,12,25,24,32,27,30);(4,19)(9,17)(13,28)(18,29)(20,21)(23,32)(25,30)(26,31)Polycyclic presentation:
2^8|a^2=e,b^2=ef,c^2=e,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=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #142402, #142403, #398734, … |
| maximal normal subgroup of | #142402, #142403, #398734, … |
| maximal quotient of | #244081, #244082, #244083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16741);