Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #1936 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1936 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
153 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 153 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(2,8,12,19)(3,14,17,22)(4,9)(5,20)(6,23)(10,25)(11,28,21,13)(16,30,26,18)(24,29)(27,31);(1,2,9,11,23,28,4,19,7,12,5,21,6,13,20,8)(3,15,22,31,30,10,26,27,17,32,14,24,18,25,16,29);(1,3,4,16,6,18,9,14,7,17,20,26,23,30,5,22)(2,10,8,24,13,15,21,27,12,25,19,31,28,32,11,29)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=d,d^2=g,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,c^b=cg,d^b=dh,e^b=eh,f^b=fh,e^c=eh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13156, #494933, #495899, … |
| maximal normal subgroup of | C2×#13156, #494933, #495899, … |
| maximal quotient of | #140189, #140190, #140211, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6144);