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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1891 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(14,17)(15,19)(21,22)(23,31)(24,32)(25,35)(26,27)(28,33)(29,34)(30,36);(5,6,8,10)(7,9,12,11)(15,19)(18,20)(22,27)(24,34)(28,31)(30,35);(5,6)(7,11)(8,10)(9,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,33)(22,28,27,31)(24,30,34,35)(25,29,36,32);(1,2,3,4)(13,15,16,19)(14,18,17,20)(21,24)(22,29)(23,30)(25,31)(26,34)(27,32)(28,36)(33,35);(5,7,8,12)(6,9,10,11)(21,25,26,36)(22,30,27,35)(23,32,33,29)(24,31,34,28)Polycyclic presentation:
2^9|b^2=h,c^2=fh,d^2=fhi,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, 4575288);