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 | |
133 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 133 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) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,7,11)(3,12,4,15)(5,10,6,9)(8,14,16,13)(17,18,22,26)(19,27,29,46)(20,31,33,45)(21,36,34,25)(23,38,39,28)(24,41,43,35)(30,47,37,42)(32,44,48,40);(1,3,7,4)(2,8,11,16)(5,14,6,13)(9,15,10,12)(17,19,22,29)(18,23,26,39)(20,32,33,48)(21,37,34,30)(24,42,43,47)(25,44,36,40)(27,45,46,31)(28,41,38,35);(1,4,7,3)(2,8,11,16)(5,13,6,14)(9,15,10,12)(20,33)(24,43)(27,46)(28,38)(31,45)(32,48)(35,41)(42,47)Polycyclic presentation:
2^8|a^2=g,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×#12413, #489147, #489149, … |
| maximal normal subgroup of | C2×#12413, #489147, #489149, … |
| maximal quotient of | #125965, #125967, #125983, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5401);