Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | Aut(G) = #72303793 (order 2048) · Inn(G) = #410 · Out(G) = #127 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #410 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×M16 | C23 |
| socle | C2 | #1466 |
| prime radical | #525 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ #405 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
167 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 167 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:
(1,2,4,9,5,10,17,25,6,7,16,20,18,21,29,32)(3,11,13,27,14,19,30,24,12,26,15,31,28,8,22,23);(2,7)(3,12)(10,21)(13,15)(14,28)(22,30)(23,24)(27,31);(1,3,5,14,6,12,18,28)(2,8,21,26,7,19,10,11)(4,15,17,22,16,13,29,30)(9,24,32,27,20,23,25,31)Polycyclic presentation:
2^8|a^2=e,c^2=fg,e^2=g,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,e^b=ef,e^c=eh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #132072, #132073, #149958, … |
| maximal normal subgroup of | #132072, #132073, #149958, … |
| maximal quotient of | #124770, #124771, #124801, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4526);