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 | #397 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
297 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 297 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(1,2,3,4)(5,6)(7,15)(8,19)(9,22)(10,14)(11,17)(12,21)(13,24)(16,32)(18,31)(20,28)(23,29)(25,33)(26,36)(27,35)(30,34);(5,7,9,17,10,18,24,36)(6,11,13,15,14,26,22,31)(8,16,23,35,21,34,20,33)(12,27,29,32,19,25,28,30);(5,8,9,23,10,21,24,20)(6,12,13,29,14,19,22,28)(7,16,17,35,18,34,36,33)(11,25,15,30,26,27,31,32)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=de,d^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^c=dh,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11350, #358608, #358836, … |
| maximal normal subgroup of | C2×#11350, #358608, #358836, … |
| maximal quotient of | #121660, #121661, #121662, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4338);