Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
260 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 260 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:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,32)(29,31);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(17,19,21,29,22,28,20,27)(18,23,24,31,25,26,30,32);(3,11)(7,15)(10,16)(12,13)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,26,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,d^2=g,e^2=g,f^2=h,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #107186, #107193, #107238, … |
| maximal normal subgroup of | #107186, #107193, #107238, … |
| maximal quotient of | #95270, #95271, #95274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1050);