Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#95 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
261 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 261 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8,6,10,5,9)(3,11,13,7,14,15,12,16)(17,18)(19,25,27,22)(20,23)(21,24)(26,31,28,30)(29,32);(1,3,5,12,6,14,4,13)(2,7,8,15,10,16,9,11)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,11,16,15)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,d^2=gh,e^2=gh,f^2=gh,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #105663, #106206, #106449, … |
| maximal normal subgroup of | #105663, #106206, #106449, … |
| maximal quotient of | #95075, #95077, #95087, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 998);