Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
870 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)(3,7)(4,5)(6,8)(10,13)(11,15)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30);(2,5)(6,7)(9,10,12,13)(11,14,15,16)(18,19)(20,27)(21,29)(30,31);(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(1,3,4,8)(2,6,5,7)(9,11,12,15)(10,14,13,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=gh,e^2=f,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 | #132544, #487349, #487901, … |
| maximal normal subgroup of | #132544, #487349, #487901, … |
| maximal quotient of | #585359, #587023, #587472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49649);