Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1866 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)(7,10)(8,9)(11,12)(15,19)(18,20)(21,22,26,27)(23,24,29,28)(25,30,33,35)(31,32,36,34);(5,7,9,12)(6,10,8,11)(14,17)(18,20)(22,27)(23,29)(25,33)(32,34);(6,8)(10,11)(13,14,16,17)(15,18,19,20)(21,23,26,29)(22,28,27,24)(25,31,33,36)(30,34,35,32);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(21,24,26,28)(22,29,27,23)(25,34,33,32)(30,31,35,36);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25)(22,30)(23,31)(24,32)(26,33)(27,35)(28,34)(29,36)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=h,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5517350);