Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #387 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1443 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
266 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 101 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:
(1,2,6,12,7,10,5,11)(3,13,15,28,16,8,21,23)(4,9,20,27,18,25,19,26)(14,29,17,32,30,22,31,24);(1,3)(2,8)(4,17,18,31)(5,15)(6,21)(7,16)(9,24,25,32)(10,13)(11,23)(12,28)(14,19,30,20)(22,26,29,27);(1,4)(2,9,10,25)(3,14)(5,20)(6,19)(7,18)(8,22,13,29)(11,27,12,26)(15,31)(16,30)(17,21)(23,32,28,24)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=d,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,g^b=gh,e^c=eh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11310, #358643, #360369, … |
| maximal normal subgroup of | C2×#11310, #358643, #360369, … |
| maximal quotient of | #121081, #121100, #121102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4298);