Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
529 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,12,10,8,19,13)(7,14,17,15,18,20,16,11);(1,2)(3,4)(5,7,10,18)(6,11,8,15)(9,16,19,17)(12,20,13,14);(2,4)(5,8)(6,10)(7,15)(9,13)(11,18)(12,19)(14,16)(17,20);(7,16)(9,19)(11,20)(12,13)(14,15)(17,18)Polycyclic presentation:
2^8|a^2=e,b^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,e^b=eh,f^c=fh,e^d=eh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #6667749, #6667759, #6667761, … |
| maximal normal subgroup of | #6667749, #6667759, #6667761, … |
| maximal quotient of | #333129, #333132, #333145, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23468);