Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8516 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
455 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 455 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,19,22,31)(12,23,25,16)(13,17)(14,18)(15,21,29,26)(20,34)(24,38,40,30)(27,32)(28,35)(33,36)(37,39);(2,5)(3,7)(9,11)(10,15)(12,20)(13,26)(14,22)(16,28)(17,31)(18,29)(19,32)(21,27)(23,37)(24,39)(25,36)(30,34)(33,38)(35,40);(9,12,14,25)(10,16,18,23)(11,20,22,36)(13,24,27,40)(15,28,29,37)(17,30,32,38)(19,33,31,34)(21,35,26,39)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,f^b=fgh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94419, #94421, #94422, … |
| maximal normal subgroup of | #94419, #94421, #94422, … |
| maximal quotient of | #97019, #97021, #97027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1512);