Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 131072 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,16)(11,15,12,14)(17,18)(19,22)(20,21)(23,24);(1,2,4,6)(3,5,8,7)(9,11,13,12)(10,14,16,15)(17,19,21,24)(18,22,20,23);(17,20,21,18)(19,22,24,23);(1,2)(3,5)(4,6)(7,8)(17,19)(18,23)(20,22)(21,24);(1,3,4,8)(2,5,6,7)(9,12,13,11)(10,15,16,14)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^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 | #98583, #251571, #251573, … |
| maximal normal subgroup of | #98583, #251571, #251573, … |
| maximal quotient of | #531660, #531686, #531875, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52247);