Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12741 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1786 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
463 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 463 conjugacy classes of subgroups — too many to draw. Only its 85 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:
(1,2,5,12,7,11,6,23)(3,13,15,29,16,9,22,28)(4,17,19,30,21,24,20,10)(8,18,26,31,25,32,27,14);(1,3)(2,8)(4,18)(5,15)(6,22)(7,16)(9,24)(10,28)(11,25)(12,26)(13,17)(14,20)(19,31)(21,32)(23,27)(29,30);(1,3)(2,9)(4,14)(5,22)(6,15)(7,16)(8,24)(10,27)(11,13)(12,29)(17,25)(18,20)(19,32)(21,31)(23,28)(26,30)Polycyclic presentation:
2^8|a^2=f,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,g^b=gh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #130776, #130779, #130782, … |
| maximal normal subgroup of | #130776, #130779, #130782, … |
| maximal quotient of | #129516, #129519, #129560, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5731);