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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31);(1,2,5,6)(3,8,7,4)(9,11,14,13)(10,15,12,16)(18,22)(19,26)(24,28)(27,32);(2,6)(3,7)(9,12,14,10)(11,15,13,16)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31);(1,3,5,7)(2,4,6,8)(9,13,14,11)(10,15,12,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,4)(2,7)(3,6)(5,8)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=g,d^2=fg,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×#54337, #8101515, #8101960, … |
| maximal normal subgroup of | C2×#54337, #8101515, #8101960, … |
| maximal quotient of | #585296, #585850, #586608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47325);