Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 65536 |
| 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 | #2617 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
917 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,8)(4,7)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,20,21)(19,23,24,22);(2,6)(4,8)(9,10,12,14)(11,13,16,15)(18,21)(22,23);(2,6)(3,7)(11,16)(13,15)(17,18)(19,22)(20,21)(23,24);(1,3,5,7)(2,4,6,8)(9,11,12,16)(10,13,14,15)(17,19,20,24)(18,22,21,23);(1,4)(2,7)(3,6)(5,8)(9,10)(11,13)(12,14)(15,16)Polycyclic presentation:
2^8|a^2=h,b^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133191, #253285, #253287, … |
| maximal normal subgroup of | #133191, #253285, #253287, … |
| maximal quotient of | #544443, #545018, #545506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34378);