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) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #392 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
437 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 437 conjugacy classes of subgroups — too many to draw. Only its 119 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)(7,17)(8,21)(9,15)(10,14)(11,16)(12,19)(13,22)(18,26)(20,30)(23,31)(24,27)(25,29)(28,35)(32,34)(33,36);(5,7,9,19,11,20,10,24)(6,12,14,17,16,27,15,30)(8,18,22,33,23,35,25,34)(13,28,21,36,29,26,31,32);(5,8)(6,13)(7,18)(9,22)(10,25)(11,23)(12,26)(14,21)(15,31)(16,29)(17,32)(19,33)(20,35)(24,34)(27,28)(30,36)Polycyclic presentation:
2^8|a^2=g,b^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,e^b=eh,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94315, #94316, #94319, … |
| maximal normal subgroup of | #94315, #94316, #94319, … |
| maximal quotient of | #116434, #116463, #116467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4099);