Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
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 | |
2078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,9,11)(7,8,12,10)(14,17)(18,20);(5,7,9,12)(6,10,11,8)(14,17)(15,19);(1,2)(3,4)(13,14)(15,18)(16,17)(19,20);(5,7)(6,8)(9,12)(10,11)(13,14)(15,18)(16,17)(19,20);(5,8)(6,7)(9,10)(11,12)(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^8|a^2=f,b^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #45439⋊C3, #72236332 |
| maximal subgroup of | #133394, #499795, #500234, … |
| maximal normal subgroup of | #133394, #499795, #500234, … |
| maximal quotient of | #557017, #557346, #557552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38427);