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 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
809 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,4,8,6)(10,13)(14,16)(18,21)(22,24);(1,3,5,8)(2,6,7,4)(10,13)(11,15);(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,20,23)(18,22,21,24);(1,4)(2,3)(5,6)(7,8)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=gh,e^2=h,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 | #97173, #172555, #172556, … |
| maximal normal subgroup of | #97173, #172555, #172556, … |
| maximal quotient of | #557185, #557186, #557514, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39278);