Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) = C24×D4×#120 (order 4096) · Inn(G) = C22×D4 · Out(G) = C24×D4 |
| 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 | C2×H(5,𝔽2) | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
611 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)(11,15)(12,14)(13,17)(16,20)(18,22)(19,21)(23,24);(10,14)(12,20)(15,21)(16,17)(19,24)(22,23);(2,5)(6,8)(9,11)(10,15)(12,19)(13,18)(14,21)(16,23)(17,22)(20,24);(11,18)(15,22)(19,24)(21,23)Polycyclic presentation:
2^8|a^2=f,e^2=g,f^2=h,b^a=be,c^a=ch,e^a=eg,e^b=eg,d^c=dg64 conjugacy classes — view the full character table.
| maximal subgroup of | #220226, #220227, #220229, … |
| maximal normal subgroup of | #220226, #220227, #220229, … |
| maximal quotient of | #160892, #160894, #160897, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10441);