Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,9)(10,12)(16,24)(19,25)(22,28)(26,27);(2,3)(13,14,17,20)(15,18,23,21)(16,25,24,19)(22,27,28,26);(5,6,8,9)(7,10,11,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,25)(26,28);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,28)(25,27);(13,16,17,24)(14,19,20,25)(15,22,23,28)(18,26,21,27)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=g,d^2=fg,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #7117001, #7117007, #7117017, … |
| maximal normal subgroup of | #7117001, #7117007, #7117017, … |
| maximal quotient of | #535406, #535554, #536064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38288);