Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | #3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(2,3)(7,8);(1,4,6,5)(2,7,3,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)Polycyclic presentation:
2^8|a^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=ch,e^a=eg,c^b=ch,e^b=eg88 conjugacy classes — view the full character table.
| maximal subgroup of | #107757, #107769, #108075, … |
| maximal normal subgroup of | #107757, #107769, #108075, … |
| maximal quotient of | #161339, #161340, #161341, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10461);