Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #391 |
| 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 | |
312 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 312 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)(3,11)(4,8,16,23)(5,17)(6,10)(7,13)(9,18)(12,26,28,24)(14,25)(15,30,31,22)(19,27,21,32)(20,29);(1,3,5,13,6,14,18,29)(2,7,9,11,10,20,17,25)(4,12,15,27,16,28,31,32)(8,21,22,26,23,19,30,24);(1,4,5,15,6,16,18,31)(2,8,9,22,10,23,17,30)(3,12,13,27,14,28,29,32)(7,19,11,24,20,21,25,26)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,d^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=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131434, #131435, #144756, … |
| maximal normal subgroup of | #131434, #131435, #144756, … |
| maximal quotient of | #120174, #120197, #120200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4233);