Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1288 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)(3,7)(4,6)(5,8)(10,11)(14,15)(19,23)(22,24);(1,2,4,6)(3,5,8,7)(10,11)(12,16)(18,21)(19,23);(9,10,13,11)(12,14,16,15)(17,18)(19,22)(20,21)(23,24);(1,2)(3,5)(4,6)(7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,22)(20,23)(21,24);(1,3)(2,5)(4,8)(6,7)(9,12,13,16)(10,14,11,15)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^8|b^2=f,c^2=g,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133530, #500065, #500413, … |
| maximal normal subgroup of | #133530, #500065, #500413, … |
| maximal quotient of | #557155, #557470, #557552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38715);