Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #124 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #400 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
265 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 265 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,14,10,15,9,16)(7,17,21,27,20,28,19,12)(8,13,24,30,25,31,23,32)(18,29,22,35,34,36,33,26);(1,2)(3,4)(5,7,11,21,10,20,9,19)(6,12,16,28,15,27,14,17)(8,22,23,18,25,33,24,34)(13,29,30,35,31,36,32,26);(2,4)(5,8,9,23,10,25,11,24)(6,13,14,30,15,31,16,32)(7,18,19,33,20,34,21,22)(12,26,17,29,27,35,28,36)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=d,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11306, #358344, #358420, … |
| maximal normal subgroup of | C2×#11306, #358344, #358420, … |
| maximal quotient of | #121079, #121096, #121098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4294);