Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C24 |
| commutator, socle | C22 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8,4,6,3,7)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24);(2,6)(7,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23);(11,18)(14,17)(19,24)(21,23);(2,6)(7,8)(10,15)(14,17)(16,22)(21,23);(1,3,4,5)(2,7,6,8)(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)Polycyclic presentation:
2^8|a^2=g,e^2=f,g^2=h,b^a=bf,d^a=df,c^b=cfh100 conjugacy classes — view the full character table.
| Fitting subgroup of | #60278⋊C3 |
| maximal subgroup of | #105698, #105703, #105938, … |
| maximal normal subgroup of | #105698, #105703, #105938, … |
| maximal quotient of | #6370259, #6370342, #6370343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53266);