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 | #2818 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,12,13)(11,14,15,16)(18,22)(19,25)(24,29)(26,31);(1,2,5,3)(4,7,8,6)(10,13)(11,15)(18,22)(23,28)(24,29)(30,32);(2,3)(6,7)(9,11,12,15)(10,14,13,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,15)(13,16)(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|a^2=f,b^2=fgh,c^2=f,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×#58334, #8104059, #8104111, … |
| maximal normal subgroup of | C2×#58334, #8104059, #8104111, … |
| maximal quotient of | #585658, #586548, #586620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51322);