Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #385 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
595 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 595 conjugacy classes of subgroups — too many to draw. Only its 175 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)(5,6,9,15,11,14,10,16)(7,17,20,30,19,12,24,26)(8,13,22,27,21,29,23,28)(18,25,33,35,32,31,34,36);(5,7,10,20,11,19,9,24)(6,12,16,26,14,17,15,30)(8,18,23,33,21,32,22,34)(13,25,28,35,29,31,27,36);(2,4)(5,8)(6,13)(7,18)(9,22)(10,23)(11,21)(12,25)(14,29)(15,27)(16,28)(17,31)(19,32)(20,33)(24,34)(26,35)(30,36)Polycyclic presentation:
2^8|a^2=f,b^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^b=eh,f^b=fh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #146607, #146608, #146618, … |
| maximal normal subgroup of | #146607, #146608, #146618, … |
| maximal quotient of | #106079, #106083, #106084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3736);