Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | C2×2−1+4 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
965 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,7,5,4,3,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24);(1,3,5,6)(2,7,4,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22);(1,3,5,6)(2,8,4,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23);(10,16)(14,17)(15,24)(22,23);(1,4,6,8,5,2,3,7)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,e^2=g,g^2=h,b^a=bf,d^a=dfh,c^b=cfh,e^b=eh88 conjugacy classes — view the full character table.
| Fitting subgroup of | #60343⋊C3 |
| maximal subgroup of | #129673, #209386, #209387, … |
| maximal normal subgroup of | #129673, #209386, #209387, … |
| maximal quotient of | #6370509, #6370511, #6370589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53331);