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 | #3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,8)(5,9)(6,10)(7,12)(13,15)(14,16)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,16,11)(19,25)(22,28)(26,31)(29,32);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,11,16,15)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=g,e^2=g,f^2=h,b^a=be,d^a=dg,e^a=eg,c^b=ch,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #107423, #107424, #107518, … |
| maximal normal subgroup of | #107423, #107424, #107518, … |
| maximal quotient of | #161463, #161467, #161472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10455);