Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7338 |
| automorphism | Aut(G) = #72286380 (order 4096) · Inn(G) = #1316 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1316 |
| commutator, prime radical | C24 | C2×C8 |
| Frattini, cosocle | #407 | C22 |
| LCS | G ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #407 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,5,10,17,24,6,8,16,22,19,23,29,31)(3,11,12,26,13,25,27,32,14,7,15,20,18,21,28,30);(1,3,6,14)(2,7,8,11)(4,15,16,12)(5,18,19,13)(9,20,22,26)(10,25,23,21)(17,27,29,28)(24,32,31,30)Polycyclic presentation:
2^8|a^2=d,b^2=h,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,d^b=de,f^b=fh,d^c=dg,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94520, #94521, #94559, … |
| maximal normal subgroup of | #94520, #94521, #94559, … |
| maximal quotient of | #93195, #93199, #93672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 329);