Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| 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 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
724 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)(3,4)(5,6)(7,8)(11,15)(14,16)(17,18,23,26)(19,21,24,20)(22,30,31,25)(27,28,32,29);(2,6)(3,7)(10,13)(14,16)(17,19,23,24)(18,20,26,21)(22,27,31,32)(25,28,30,29);(9,10)(11,14)(12,13)(15,16)(17,20,23,21)(18,24,26,19)(22,28,31,29)(25,32,30,27);(1,3,5,7)(2,4,6,8)(9,11)(10,14)(12,15)(13,16)(17,21,23,20)(18,19,26,24)(22,29,31,28)(25,27,30,32);(1,4)(2,7)(3,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| maximal subgroup of | #98114, #179259, #179268, … |
| maximal normal subgroup of | #98114, #179259, #179268, … |
| maximal quotient of | #606736, #606793, #606797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46207);