Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C24 |
| commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3613 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)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24);(10,15)(11,17)(16,22)(18,23);(10,15)(14,20)(16,22)(21,24);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ch88 conjugacy classes — view the full character table.
| Fitting subgroup of | #62686⋊C3 |
| maximal subgroup of | #184908, #184910, #184916, … |
| maximal normal subgroup of | #184908, #184910, #184916, … |
| maximal quotient of | #7625282, #7625309, C4×#3459, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55674);