Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C22 | C22×C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
791 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,8,7)(9,10,12,11)(13,14,17,15)(16,19,20,18);(9,10,12,11)(14,15)(18,19);(1,2)(3,4)(5,7,8,6)(9,11,12,10)(13,15,17,14)(16,19,20,18);(13,16,17,20)(14,18,15,19)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=ce88 conjugacy classes — view the full character table.
| maximal subgroup of | #115917, #115918, #120124, … |
| maximal normal subgroup of | #115917, #115918, #120124, … |
| maximal quotient of | #153718, #153722, #153774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6771);