Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 262144 |
| 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×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
943 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,8)(17,18)(19,21)(20,23)(22,24);(2,5)(3,7)(9,10,13,16)(11,15,12,14)(18,19)(22,23);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,16,15)(17,19,21,18)(20,23,24,22);(17,20,21,24)(18,22,19,23);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)Polycyclic presentation:
2^8|b^2=gh,c^2=fgh,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #98417, #258627, #258629, … |
| maximal normal subgroup of | #98417, #258627, #258629, … |
| maximal quotient of | #521500, #522202, #522757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32622);