Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12707 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1781 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | #3487 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
501 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 501 conjugacy classes of subgroups — too many to draw. Only its 73 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)(3,11,14,25)(4,8)(5,17,20,9)(6,10)(7,13)(12,28,30,27)(15,26,19,23)(16,24)(18,22)(21,29)(31,32);(1,3)(2,7)(4,12)(5,18)(6,14)(8,21)(9,25)(10,22)(11,17)(13,20)(15,31)(16,30)(19,29)(23,27)(24,32)(26,28);(1,4,6,16)(2,8,10,24)(3,12,14,30)(5,19,20,15)(7,21,22,32)(9,26,17,23)(11,27,25,28)(13,31,18,29)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #133841, #135531, #139736, … |
| maximal normal subgroup of | #133841, #135531, #139736, … |
| maximal quotient of | #129223, #129259, #129347, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5696);