Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1391 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(6,10)(8,15)(11,17)(12,16)(14,19)(18,20);(1,2,3,4);(5,6,9,12)(7,11,13,18)(8,16,15,10)(14,20,19,17);(5,7)(6,11)(8,14)(9,13)(10,17)(12,18)(15,19)(16,20);(7,13)(11,18)(14,19)(17,20)Polycyclic presentation:
2^8|b^2=f,c^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,g^c=gh,e^d=eh61 conjugacy classes — view the full character table.
| Fitting subgroup of | #60431⋊C3 |
| maximal subgroup of | #245381, #245383, #245395, … |
| maximal normal subgroup of | #245381, #245383, #245395, … |
| maximal quotient of | #6356017, #6356018, #6356019, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53419);