Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12833 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #1807 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1807 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
179 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 179 conjugacy classes of subgroups — too many to draw. Only its 77 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)(3,14,18,32)(4,19,21,29)(5,15)(6,11)(7,13)(8,16,22,9)(10,23)(12,25)(17,31,26,20)(24,27)(28,30);(1,3,5,16,6,17,23,29,7,18,24,9,25,26,30,19)(2,8,10,14,12,4,15,20,13,22,28,32,11,21,27,31);(1,4)(2,9)(3,15)(5,20)(6,22)(7,21)(8,25)(10,26)(11,29)(12,19)(13,16)(14,30)(17,28)(18,27)(23,32)(24,31)Polycyclic presentation:
2^8|a^2=f,b^2=d,d^2=g,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,f^b=fh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12833, #493807, #495899, … |
| maximal normal subgroup of | C2×#12833, #493807, #495899, … |
| maximal quotient of | #134661, #134692, #134745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5821);