Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #390 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1439 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
434 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 434 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)(4,15,17,8)(5,18)(6,10)(7,13)(9,19)(12,24,27,32)(14,25)(16,23,30,29)(20,31,22,26)(21,28);(1,3,5,13,6,14,19,28)(2,7,9,11,10,21,18,25)(4,12,16,26,17,27,30,31)(8,22,23,32,15,20,29,24);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,23)(10,15)(11,24)(13,26)(14,27)(18,29)(19,30)(21,22)(25,32)(28,31)Polycyclic presentation:
2^8|a^2=g,b^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,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131350, #131351, #144726, … |
| maximal normal subgroup of | #131350, #131351, #144726, … |
| maximal quotient of | #116429, #116448, #116452, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4090);