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×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
895 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,7)(10,14)(11,12)(17,18,21,19)(20,22,24,23);(1,2,5,4)(3,6,8,7)(9,10,13,14)(11,15,12,16)(20,24)(22,23);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,16)(13,14)(17,19,21,18)(20,23,24,22);(9,11,13,12)(10,15,14,16)(17,20)(18,22)(19,23)(21,24);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,16,14,15)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fgh,d^2=g,e^2=f,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 | #7085034, #7085048, #7085097, … |
| maximal normal subgroup of | #7085034, #7085048, #7085097, … |
| maximal quotient of | #532045, #532129, #532248, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52685);