Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2673 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
833 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(4,7)(17,18,23,24)(19,20,21,22);(1,2)(3,7)(4,8)(5,6)(9,10,13,15)(11,12,16,14)(17,19,23,21)(18,20,24,22);(2,6)(3,8)(9,11,13,16)(10,14,15,12)(17,20,23,22)(18,19,24,21);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,21,23,19)(18,20,24,22);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,22,23,20)(18,19,24,21)Polycyclic presentation:
2^8|a^2=g,b^2=fgh,c^2=fgh,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#57572, #8100997, #8101185, … |
| maximal normal subgroup of | C2×#57572, #8100997, #8101185, … |
| maximal quotient of | #585488, #586478, #586520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50560);