Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | Aut(G) = #72313530 (order 4096) · Inn(G) = #381 · Out(G) = #379 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #381 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1817 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
208 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 81 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:
(1,2,4,9,5,10,16,23,6,7,15,18,17,19,28,29)(3,11,13,24,14,25,27,32,12,8,20,21,26,22,30,31);(2,7)(4,15)(8,11)(10,19)(13,20)(16,28)(22,25)(27,30);(1,3)(2,8,7,11)(4,13)(5,14)(6,12)(9,21,18,24)(10,22,19,25)(15,20)(16,27)(17,26)(23,31,29,32)(28,30)Polycyclic presentation:
2^8|a^2=f,c^2=d,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^b=fh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #129304, #129306, #129309, … |
| maximal normal subgroup of | #129304, #129306, #129309, … |
| maximal quotient of | #135159, #135160, #135190, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5839);