Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1366 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #333 |
| commutator | C42 | C42 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C2 | #1296 |
| prime radical | C23 | #101 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,6,8,18,11)(3,12,14,7,16,21,13,23)(5,19,17,10,20,25,28,24)(15,22,29,31,30,32,27,26);(1,3,5,15,4,13,17,27,6,16,20,30,18,14,28,29)(2,7,10,22,11,21,19,31,8,23,24,32,9,12,25,26)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=h,e^2=fgh,g^2=h,b^a=bc,c^a=cf,e^a=efgh,g^a=gh,c^b=cg,d^b=dh,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #94686, #94687, C2×#7273, … |
| maximal normal subgroup of | #94686, #94687, C2×#7273, … |
| maximal quotient of | #93093, #93094, #93097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 261);