Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
1637 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)(11,16)(12,17)(14,19)(18,20);(5,6,8,10,9,11,15,16)(7,12,14,17,13,18,19,20);(1,2,3,4)(6,11)(10,16)(12,18)(17,20);(5,7)(6,12)(8,14)(9,13)(10,17)(11,18)(15,19)(16,20);(7,13)(12,18)(14,19)(17,20)Polycyclic presentation:
2^8|b^2=f,c^2=g,f^2=h,b^a=bf,c^a=cg,f^a=fh,c^b=ch,e^d=eh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #60650⋊C3 |
| maximal subgroup of | #152051, #249799, #249800, … |
| maximal normal subgroup of | #152051, #249799, #249800, … |
| maximal quotient of | #6357223, #6357224, #6357226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53638);