Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12173 |
| automorphism | Aut(G) = #72308726 (order 2048) · Inn(G) = #1598 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1598 |
| commutator, Frattini, cosocle | C2×M16 | C23 |
| prime radical | #575 | C22 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
170 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 35 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)(4,18)(5,9)(6,11)(7,12)(8,14)(10,20)(15,26)(16,29)(17,30)(19,23)(21,27)(22,28)(24,32)(25,31);(1,3,7,17)(2,8,12,25)(4,19,21,32)(5,16,6,15)(9,23,11,24)(10,26,28,29)(13,27,30,18)(14,22,31,20);(2,9,12,11)(8,23,25,24)(10,27,28,18)(13,26,30,29)Polycyclic presentation:
2^8|b^2=h,c^2=g,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gh,d^b=df,e^b=egh,f^b=fh,d^c=dg,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | #151167, #151169, #151253, … |
| maximal normal subgroup of | #151167, #151169, #151253, … |
| maximal quotient of | #123132, #123134, #123140, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5162);