Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #387 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1441 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
310 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,13)(4,17,20,27)(5,11)(6,10)(7,12)(8,15)(9,19,28,22)(14,23,30,32)(16,24)(18,26,29,31)(21,25);(1,3,5,15,7,16,6,21)(2,8,10,13,12,25,11,24)(4,18,19,14,20,29,22,30)(9,26,27,32,28,31,17,23);(1,4,5,19,7,20,6,22)(2,9,10,27,12,28,11,17)(3,14,15,29,16,30,21,18)(8,23,13,26,25,32,24,31)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11247, #358643, #360226, … |
| maximal normal subgroup of | C2×#11247, #358643, #360226, … |
| maximal quotient of | #120175, #120201, #120203, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4235);