Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | Aut(G) = #72303840 (order 2048) · Inn(G) = #410 · Out(G) = C25 |
| 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 | #1467 |
| prime radical | C23 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ #405 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,11,6,10,21,27,7,12,19,29,4,8,18,23)(3,13,15,30,16,26,31,28,17,22,20,32,14,9,24,25);(2,8,12,10)(5,19)(9,22,26,13)(11,23,29,27)(15,20)(18,21)(24,31)(25,32,28,30);(1,3,6,16,7,17,4,14)(2,9,8,22,12,26,10,13)(5,20,21,24,19,15,18,31)(11,28,23,30,29,25,27,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=fg,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,e^b=efh,e^c=eh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94265, #94266, #94267, … |
| maximal normal subgroup of | #94265, #94266, #94267, … |
| maximal quotient of | #124774, #124775, #124805, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4527);