Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13340 |
| automorphism | Aut(G) = #72317053 (order 4096) · Inn(G) = #493 · Out(G) = #392 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #493 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2011 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
276 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 59 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)(4,11)(5,10)(6,24)(7,12)(8,15)(9,16)(13,20)(17,26)(18,32)(19,29)(21,31)(22,27)(23,30)(25,28);(1,3)(2,8)(4,19)(5,15)(6,25)(7,18)(9,24)(10,28)(11,14)(12,27)(13,29)(16,21)(17,20)(22,23)(26,30)(31,32);(1,3)(2,9)(4,16)(5,22)(6,25)(7,18)(8,13)(10,14)(11,26)(12,29)(15,23)(17,20)(19,21)(24,27)(28,31)(30,32)Polycyclic presentation:
2^8|d^2=gh,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=eg,g^b=gh,d^c=dgh,e^c=egh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #125642, #125643, #125729, … |
| maximal normal subgroup of | #125642, #125643, #125729, … |
| maximal quotient of | #142644, #142645, #142646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6328);