Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
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 | |
3793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(6,8)(12,16)(14,15);(5,6)(7,8)(10,11)(14,15);(1,2)(3,4)(5,6)(7,8)(9,10,13,11)(12,14,16,15);(9,11,13,10)(12,15,16,14);(9,12)(10,14)(11,15)(13,16)Polycyclic presentation:
2^8|c^2=h,d^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| Frattini subgroup of | #72238704, #72238705 |
| maximal subgroup of | #111945, #113037, #113640, … |
| maximal normal subgroup of | #111945, #113037, #113640, … |
| maximal quotient of | #516388, #516399, #516581, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28210);