Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C25×C4×D4 (order 1024) · Inn(G) = C2×D4 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C8×D4 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
235 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 235 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,16,11,17,30,21,12,18,31,35,9,15,20,33)(7,19,14,34,22,8,24,26,23,25,28,36,27,13,32,29);(1,2,3,4)(5,7,20,24,9,27,31,14,12,23,30,32,11,22,10,28)(6,13,33,34,15,25,35,29,18,8,21,36,17,19,16,26);(5,8)(6,14)(7,21)(9,19)(10,29)(11,25)(12,13)(15,32)(16,27)(17,24)(18,28)(20,36)(22,35)(23,33)(26,31)(30,34)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12980, #494412, #494413, … |
| maximal normal subgroup of | C2×#12980, #494412, #494413, … |
| maximal quotient of | #136575, #136583, #136591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5968);