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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
794 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)(3,4)(5,6)(7,8)(9,10,12,15)(11,14,16,13)(17,18,21,23)(19,20,24,22);(2,6)(3,7)(9,11,12,16)(10,13,15,14)(17,19)(18,22)(20,23)(21,24);(11,16)(13,14)(17,20,21,22)(18,19,23,24);(1,3,5,7)(2,4,6,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,19)(21,22)(23,24);(1,4)(2,7)(3,6)(5,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,20)(21,23)(22,24)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=g,d^2=fgh,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×#58397, #8104246, #8104424, … |
| maximal normal subgroup of | C2×#58397, #8104246, #8104424, … |
| maximal quotient of | #585636, #585714, #586160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51385);