Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #390 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1445 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
290 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 290 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)(3,11)(4,15,17,8)(5,18)(6,10)(7,13)(9,20)(12,26,28,32)(14,25)(16,24,31,29)(19,22)(21,30,23,27);(1,3,5,19,6,14,20,13)(2,7,18,11,10,22,9,25)(4,12,16,30,17,28,31,27)(8,23,29,32,15,21,24,26);(1,4,5,16,6,17,20,31)(2,8,18,29,10,15,9,24)(3,12,19,30,14,28,13,27)(7,21,11,26,22,23,25,32)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=de,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11356, #358715, #362913, … |
| maximal normal subgroup of | C2×#11356, #358715, #362913, … |
| maximal quotient of | #121641, #121642, #121643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4344);