Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| 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×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1078 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)(10,13)(14,16)(18,21)(19,23);(1,2,4,6)(3,5,8,7)(9,10)(11,14)(12,13)(15,16)(17,18,20,21)(19,22,23,24);(17,18)(19,22)(20,21)(23,24);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,14,13,16)(17,18)(19,22)(20,21)(23,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^8|b^2=h,d^2=fh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #130318, #485953, #486399, … |
| maximal normal subgroup of | #130318, #485953, #486399, … |
| maximal quotient of | #531877, #532139, #532507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52902);