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 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
332 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 332 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:
(1,2,5,9,6,10,12,21)(3,11,14,23,15,26,20,7)(4,16,19,30,18,8,24,25)(13,27,28,29,17,31,32,22);(1,3,12,14,6,15,5,20)(2,7,21,11,10,23,9,26)(4,17,24,32,18,13,19,28)(8,22,30,27,16,29,25,31);(1,4,5,19,6,18,12,24)(2,8,9,25,10,16,21,30)(3,13,14,28,15,17,20,32)(7,22,11,27,23,29,26,31);(4,18)(8,16)(13,17)(19,24)(22,29)(25,30)(27,31)(28,32)Polycyclic presentation:
2^8|a^2=e,b^2=ef,c^2=e,e^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,f^b=fh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #142400, #142401, #396791, … |
| maximal normal subgroup of | #142400, #142401, #396791, … |
| maximal quotient of | #244081, #244082, #244083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16739);