Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#99 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C2 | C2×#342 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
368 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 368 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:
(1,2,5,10,7,12,6,11)(3,13,16,29,14,28,17,30)(4,18,19,27,20,9,21,25)(8,22,24,15,23,32,26,31);(1,3,6,17,7,14,5,16)(2,8,11,24,12,23,10,26)(4,15,19,32,20,31,21,22)(9,13,25,30,18,28,27,29);(3,14)(8,23)(13,28)(15,31)(16,17)(22,32)(24,26)(29,30)Polycyclic presentation:
2^8|a^2=e,b^2=f,e^2=h,f^2=h,b^a=bd,d^a=dg,g^a=gh,c^b=ch,e^b=eg,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #146478, #146489, #146496, … |
| maximal normal subgroup of | #146478, #146489, #146496, … |
| maximal quotient of | #126912, #126950, #126976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5567);