Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2032 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #393 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1439 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
426 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 426 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:
(1,2,5,10,6,9,19,24)(3,7,13,22,12,21,28,31)(4,14,17,25,16,8,18,23)(11,26,27,32,15,20,29,30);(1,3)(2,7)(4,15,16,11)(5,13)(6,12)(8,20,14,26)(9,21)(10,22)(17,29,18,27)(19,28)(23,30,25,32)(24,31);(1,4)(2,8)(3,11)(5,18)(6,16)(7,20)(9,14)(10,25)(12,15)(13,29)(17,19)(21,26)(22,32)(23,24)(27,28)(30,31)Polycyclic presentation:
2^8|a^2=g,b^2=d,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fh,c^b=cf,f^b=fh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11090, #359177, #359789, … |
| maximal normal subgroup of | C2×#11090, #359177, #359789, … |
| maximal quotient of | #115977, #115996, #115997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4078);