Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13077 |
| automorphism | Aut(G) = #72314018 (order 4096) · Inn(G) = #1893 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1893 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3060 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
235 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 235 conjugacy classes of subgroups — too many to draw. Only its 73 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)(3,13,16,28)(4,9,19,27)(5,20)(6,12)(7,25)(8,15,26,21)(10,14)(11,24)(17,18,31,22)(23,32)(29,30);(1,3,14,27,24,31,5,21,6,16,30,9,7,17,23,15)(2,8,20,18,25,4,10,28,12,26,32,22,11,19,29,13);(1,4)(2,9)(3,10)(5,22)(6,19)(7,8)(11,21)(12,27)(13,30)(14,28)(15,25)(16,29)(17,20)(18,23)(24,26)(31,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #130232, #130233, #130288, … |
| maximal normal subgroup of | #130232, #130233, #130288, … |
| maximal quotient of | #137621, #137622, #137661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6065);