Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11994 |
| automorphism | Aut(G) = #72306881 (order 2048) · Inn(G) = #1558 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1558 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | C23 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9,18,10,28,27,6,11,17,26,5,8,19,24)(3,12,14,25,20,23,16,31,13,29,30,32,15,7,21,22);(1,3,6,13)(2,7)(4,16,17,21)(5,20,18,15)(8,12)(9,25)(10,29)(11,23)(14,19,30,28)(22,27)(24,31)(26,32);(4,17)(9,26)(14,30)(16,21)(19,28)(22,31)(24,27)(25,32)Polycyclic presentation:
2^8|a^2=e,b^2=g,d^2=gh,e^2=fg,f^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efg,f^b=fh,e^c=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #133271, #133272, #133273, … |
| maximal normal subgroup of | #133271, #133272, #133273, … |
| maximal quotient of | #124558, #124559, #124647, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4982);