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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #393 |
| commutator, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1442 |
| LCS | G ▷ C24 ▷ 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:
(1,2,5,10,6,9,20,26)(3,11,14,27,13,7,19,22)(4,15,18,29,17,8,23,25)(12,21,28,31,16,24,30,32);(1,3)(2,7)(4,16,17,12)(5,19)(6,13)(8,24,15,21)(9,11)(10,27)(14,20)(18,28,23,30)(22,26)(25,31,29,32);(1,4)(2,8,9,15)(3,12)(5,18)(6,17)(7,21,11,24)(10,25,26,29)(13,16)(14,28)(19,30)(20,23)(22,31,27,32)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fh,g^b=gh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11280, #359177, #360371, … |
| maximal normal subgroup of | C2×#11280, #359177, #360371, … |
| maximal quotient of | #120635, #120727, #120730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4268);