Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #388 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1442 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
288 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 288 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)(3,13)(4,18,22,26)(5,11)(6,10)(7,12)(8,15)(9,20,27,21)(14,29,31,23)(16,24)(17,28)(19,32,30,25);(1,3,5,16,7,17,6,15)(2,8,11,13,12,24,10,28)(4,19,20,31,22,30,21,14)(9,25,18,29,27,32,26,23);(1,4,5,20,7,22,6,21)(2,9,10,26,12,27,11,18)(3,14,15,30,17,31,16,19)(8,23,13,25,24,29,28,32)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,f^b=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11265, #358422, #360827, … |
| maximal normal subgroup of | C2×#11265, #358422, #360827, … |
| maximal quotient of | #120689, #120690, #120711, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4253);