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 | |
992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,24)(17,21)(19,23)(20,26)(22,28)(25,27);(6,9)(10,12)(13,15,17,23)(14,18,21,19)(16,25,26,22)(20,28,24,27);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,28)(24,27)(25,26);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,20)(17,21)(19,23)(22,27)(24,26)(25,28);(13,16,17,26)(14,20,21,24)(15,22,23,25)(18,27,19,28)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=f,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 | #6999001, #6999005, #6999065, … |
| maximal normal subgroup of | #6999001, #6999005, #6999065, … |
| maximal quotient of | #535312, #535580, #535940, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38314);