Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C23 | C23×C4 |
| Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(9,10,11,12)(13,14)(15,19)(16,18)(17,20);(1,2,4,3)(5,6)(7,8)(13,15,17,16)(14,18,20,19);(1,3,4,2)(6,8)(13,15,17,16)(14,19,20,18);(13,16,17,15)(14,18,20,19)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=e,d^2=ef,b^a=be,c^a=cf,c^b=cg88 conjugacy classes — view the full character table.
| maximal subgroup of | #109922, #109923, #116330, … |
| maximal normal subgroup of | #109922, #109923, #116330, … |
| maximal quotient of | #153736, #153739, C4×#1441, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6836);