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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,7)(5,6)(10,13)(11,15)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31);(1,3)(2,7)(4,6)(5,8)(9,10,12,13)(11,16,15,14)(18,22)(19,26)(24,28)(27,32);(3,4)(7,8)(9,10,12,13)(11,14,15,16)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31);(1,4,6,3)(2,7,5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,5,6,2)(3,7,4,8)(9,11)(10,14)(12,15)(13,16)(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=fh,c^2=g,d^2=fg,e^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×#56236, #8101780, #8101948, … |
| maximal normal subgroup of | C2×#56236, #8101780, #8101948, … |
| maximal quotient of | #585470, #585960, #586720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49224);