Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
759 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(9,10,11,12)(13,14)(15,19)(16,20)(17,18);(1,2,4,3)(6,8)(14,18)(15,20);(5,6,7,8)(9,10)(11,12)(13,15,17,20)(14,19,18,16);(1,3,4,2)(13,16,17,19)(14,20,18,15)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#15074, #608191, #608196, … |
| maximal normal subgroup of | C2×#15074, #608191, #608196, … |
| maximal quotient of | #154156, #154253, #154274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8062);