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 | |
465 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 465 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)(3,12,14,8)(4,7)(5,19)(6,10)(9,16)(11,17)(13,25,30,20)(15,24,29,27)(18,21)(22,31)(23,28,26,32);(2,7)(4,16)(5,17)(8,20)(9,11)(10,21)(12,25)(13,28)(15,29)(18,31)(19,22)(23,24)(26,27)(30,32);(1,3)(2,8)(4,13)(5,15)(6,14)(7,20)(9,23)(10,12)(11,24)(16,28)(17,29)(18,30)(19,27)(21,25)(22,26)(31,32)Polycyclic presentation:
2^8|a^2=f,d^2=gh,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, #130780, … |
| maximal normal subgroup of | #130776, #130779, #130780, … |
| maximal quotient of | #129516, #129517, #129559, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5729);