Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #399 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
347 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 127 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:
(5,6,10,14,11,16,9,15)(7,17,21,27,20,28,19,12)(8,13,23,31,25,32,24,30)(18,29,34,26,35,33,22,36);(1,2)(3,4)(5,7,9,19,11,20,10,21)(6,12,14,17,16,27,15,28)(8,22,23,18,25,34,24,35)(13,29,30,36,32,33,31,26);(2,4)(5,8)(6,13)(7,18,20,35)(9,24)(10,23)(11,25)(12,26,27,36)(14,31)(15,30)(16,32)(17,33,28,29)(19,22,21,34)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=e,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93963, #136141, #136265, … |
| maximal normal subgroup of | #93963, #136141, #136265, … |
| maximal quotient of | #121338, #121339, #121340, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4324);