Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #397 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1445 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
280 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 109 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:
(1,2,6,12,7,10,5,11)(3,8,16,24,14,25,15,23)(4,17,19,27,21,28,20,9)(13,26,30,31,29,32,18,22);(1,3,5,16,7,14,6,15)(2,8,12,23,10,25,11,24)(4,18,19,29,21,30,20,13)(9,26,28,31,27,32,17,22);(1,4,5,19,7,21,6,20)(2,9,12,28,10,27,11,17)(3,13,16,18,14,29,15,30)(8,22,23,26,25,31,24,32)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=de,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11363, #359281, #361946, … |
| maximal normal subgroup of | C2×#11363, #359281, #361946, … |
| maximal quotient of | #121706, #121709, #121710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4351);