Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 | |
479 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,11,16)(7,14,8,18)(9,17,13,20)(10,19,12,15);(1,3,4,2)(5,7,12,20,11,8,10,17)(6,13,19,18,16,9,15,14);(2,3)(5,8)(6,14)(7,11)(9,15)(10,20)(12,17)(13,19)(16,18);(5,9,10,18,11,13,12,14)(6,7,15,17,16,8,19,20)Polycyclic presentation:
2^8|a^2=f,b^2=ef,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25436, #6444298, #6444299, … |
| maximal normal subgroup of | C2×#25436, #6444298, #6444299, … |
| maximal quotient of | #288107, #288112, #288115, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18424);