Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13512 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #2085 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2085 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
116 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 51 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:
(1,2,7,11)(3,12,9,5)(4,13)(6,21,23,10)(8,18)(14,20,28,29)(15,24)(16,27)(17,30)(19,25)(22,32)(26,31);(1,3,5,2,7,9,12,11)(4,15,16,26,19,30,18,32)(6,21,20,14,23,10,29,28)(8,24,25,22,27,17,13,31);(1,4,5,16,7,19,12,18)(2,8,9,25,11,27,3,13)(6,22,20,17,23,31,29,24)(10,30,28,32,21,15,14,26)Polycyclic presentation:
2^8|a^2=eg,b^2=d,c^2=d,d^2=h,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=eh,f^b=fg,g^b=gh,f^c=fg,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13512, #499416, #499603, … |
| maximal normal subgroup of | C2×#13512, #499416, #499603, … |
| maximal quotient of | #146171, #146177, #146180, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6500);