Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
623 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24);(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(2,5)(6,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18);(10,17)(15,24)(16,21)(18,23)Polycyclic presentation:
2^8|a^2=f,c^2=g,e^2=g,f^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #105656, #105663, #105926, … |
| maximal normal subgroup of | #105656, #105663, #105926, … |
| maximal quotient of | #160782, #160787, #160790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10438);