Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C25 |
| commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,19)(13,22)(14,17)(16,20)(18,24)(21,23);(2,5)(6,8)(9,11,14,13)(10,15,17,22)(12,18,20,21)(16,23,19,24);(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,19)(22,24);(12,20)(16,19)(18,21)(23,24);(9,13,14,11)(10,15,17,22)(12,21,20,18)(16,23,19,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|b^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,d^c=dh88 conjugacy classes — view the full character table.
| Fitting subgroup of | #62810⋊C3 |
| maximal subgroup of | #186966, #186975, #187002, … |
| maximal normal subgroup of | #186966, #186975, #187002, … |
| maximal quotient of | #7625441, #7625463, #7625466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55798);