Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | |
979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,14)(11,12,16,15)(17,18,23,21)(19,24,22,20);(2,3)(6,7)(10,14)(12,15)(17,19,23,22)(18,24,21,20);(9,11)(10,15)(12,14)(13,16)(17,20)(18,22)(19,21)(23,24);(1,3,5,2)(4,7,8,6)(9,12,13,15)(10,11,14,16)(17,21,23,18)(19,24,22,20);(1,4,5,8)(2,6,3,7)(9,11)(10,15)(12,14)(13,16)(17,22,23,19)(18,24,21,20)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #98074, #167993, #167994, … |
| maximal normal subgroup of | #98074, #167993, #167994, … |
| maximal quotient of | #553868, #553873, #554105, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30205);