Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(10,13)(14,15)(18,22)(19,25)(24,29)(26,31);(1,2,5,3)(4,7,8,6)(9,10,12,13)(11,15,16,14)(18,22)(23,28)(24,29)(30,32);(2,3)(6,7)(9,11,12,16)(10,14,13,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)Polycyclic presentation:
2^8|b^2=gh,c^2=fg,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#58360, #8104034, #8104120, … |
| maximal normal subgroup of | C2×#58360, #8104034, #8104120, … |
| maximal quotient of | #586034, #586692, #587164, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51348);