Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,7,8)(10,13)(11,15)(17,18)(19,20)(21,23)(22,24)(25,26,29,27)(28,30,32,31);(2,6)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,21)(18,23,24,20)(26,27)(30,31);(2,6)(4,8)(11,15)(14,16)(18,24)(19,21)(25,27,29,26)(28,30,32,31);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,23)(18,19,24,21);(1,4)(2,3)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(17,21,22,19)(18,23,24,20)(25,28)(26,30)(27,31)(29,32)Polycyclic presentation:
2^9|a^2=gh,b^2=fghi,c^2=fgi,d^2=fghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4637402);