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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
967 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)(6,9)(10,12)(13,14,18,21)(15,16,19,24)(17,25,27,20)(22,28,26,23);(2,4)(13,15,18,19)(14,16,21,24)(17,26,27,22)(20,28,25,23);(5,6,8,9)(7,10,11,12)(13,15)(14,16)(17,22)(18,19)(20,23)(21,24)(25,28)(26,27);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,25)(27,28);(13,17,18,27)(14,20,21,25)(15,22,19,26)(16,23,24,28)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=g,d^2=g,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 | #7002066, #7002070, #7002073, … |
| maximal normal subgroup of | #7002066, #7002070, #7002073, … |
| maximal quotient of | #535320, #535534, #535959, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38331);