Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
633 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,7,8)(10,12)(14,15)(17,18,22,20)(19,24,21,23);(5,6)(7,8)(9,10,13,12)(11,15,16,14)(17,19,22,21)(18,23,20,24);(5,6)(7,8)(9,11,13,16)(10,14,12,15)(17,20,22,18)(19,24,21,23);(1,2,3,4)(9,12,13,10)(11,15,16,14)(17,21,22,19)(18,24,20,23)Polycyclic presentation:
2^8|a^2=eg,b^2=ef,c^2=ef,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #103250, #103266, #103282, … |
| maximal normal subgroup of | #103250, #103266, #103282, … |
| maximal quotient of | #154175, #154388, #154389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8144);