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 | |
3595 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)(12,20)(13,17)(14,18)(16,21)(19,24)(22,23);(2,5)(6,8)(9,11,14,13)(10,15,18,17)(12,19,21,22)(16,23,20,24);(9,12,14,21)(10,16,18,20)(11,19,13,22)(15,23,17,24);(9,13,14,11)(10,17,18,15)(12,19,21,22)(16,23,20,24);(10,18)(15,17)(16,20)(23,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^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 | #62793⋊C3 |
| maximal subgroup of | #184954, #184958, #184962, … |
| maximal normal subgroup of | #184954, #184958, #184962, … |
| maximal quotient of | #7625433, #7625458, #7625506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55781);