Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12705 |
| automorphism | Aut(G) = #72340430 (order 4096) · Inn(G) = #1781 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1781 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | #3485 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
500 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 500 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)(4,8)(5,17,19,9)(6,10)(7,13,21,31)(12,27)(14,26)(15,24,18,22)(16,23)(20,28,32,30)(25,29);(1,3)(2,7)(4,12)(5,13)(6,14)(8,20)(9,11)(10,21)(15,28)(16,29)(17,26)(18,30)(19,31)(22,27)(23,32)(24,25);(1,4,6,16)(2,8,10,23)(3,12,14,29)(5,18,19,15)(7,20,21,32)(9,24,17,22)(11,25,26,27)(13,30,31,28)Polycyclic presentation:
2^8|a^2=f,c^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,e^b=eh,f^b=fg,d^c=dh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #133840, #135517, #139730, … |
| maximal normal subgroup of | #133840, #135517, #139730, … |
| maximal quotient of | #129221, #129260, #129331, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5694);