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) = C2×C42 |
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×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
302 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 302 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)(3,9)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16)(17,18)(19,25,27,22)(20,23)(21,24)(26,31,28,30)(29,32);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32);(3,10)(6,14)(9,15)(11,16)(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=h,d^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 | #105654, #105655, #106195, … |
| maximal normal subgroup of | #105654, #105655, #106195, … |
| maximal quotient of | #94961, #94964, #94973, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 933);