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) = C27 |
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 | |
2186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(7,11)(10,12)(14,17)(18,20)(21,22)(23,24)(25,28)(26,29)(27,30)(31,32)(33,34)(35,36);(6,9)(10,12)(15,19)(18,20)(21,23,27,25)(22,28,30,24)(26,31,35,33)(29,34,36,32);(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,24,27,28)(22,23,30,25)(26,34,35,32)(29,33,36,31);(1,2)(3,4)(13,15)(14,18)(16,19)(17,20)(21,25,27,23)(22,28,30,24)(26,33,35,31)(29,34,36,32);(5,7)(6,10)(8,11)(9,12)(21,26,27,35)(22,29,30,36)(23,31,25,33)(24,32,28,34)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=f,e^2=f,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, 4514670);