Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | Aut(G) = C23×#21452 (order 2048) · Inn(G) = #447 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #446 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
136 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,11)(3,12,4,15)(5,10,6,9)(8,14,16,13)(18,22)(19,25)(20,28)(23,31)(24,32)(26,29)(27,30);(1,3,7,4)(2,8,11,16)(5,14,6,13)(9,15,10,12)(17,18,21,23)(19,26,27,32)(20,29,28,24)(22,30,31,25);(1,4,7,3)(2,8,11,16)(5,13,6,14)(9,15,10,12)(19,27)(22,31)(25,30)(26,32)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=gh,d^2=fh,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=egh,f^a=fh,d^b=df,e^b=eg,f^b=fh,d^c=dg,e^c=egh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12415, #489147, #489149, … |
| maximal normal subgroup of | C2×#12415, #489147, #489149, … |
| maximal quotient of | #125961, #125963, #125980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5403);