Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(6,7)(9,10)(11,15)(12,13)(14,16);(2,4)(6,7)(10,13)(14,15);(5,6,8,7);(1,2)(3,4)(5,7,8,6)(9,10)(11,14)(12,13)(15,16);(9,11,12,16)(10,14,13,15)Polycyclic presentation:
2^8|c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #98619, #246439, #246440, … |
| maximal normal subgroup of | #98619, #246439, #246440, … |
| maximal quotient of | #531684, #531695, #531766, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52112);