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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2135 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)(6,9)(10,12)(13,14,16,18)(15,17,19,20)(21,22,26,27)(23,25,31,28)(24,32,34,29)(30,33,36,35);(7,11)(10,12)(13,15,16,19)(14,17,18,20)(22,27)(25,28)(29,32)(33,35);(5,6,8,9)(7,10,11,12)(13,15)(14,17)(16,19)(18,20)(21,23)(22,28)(24,30)(25,27)(26,31)(29,33)(32,35)(34,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(21,24,26,34)(22,29,27,32)(23,30,31,36)(25,35,28,33);(5,7,8,11)(6,10,9,12)(13,14)(15,17)(16,18)(19,20)(21,25,26,28)(22,23,27,31)(24,33,34,35)(29,36,32,30)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=fg,d^2=f,e^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, 5516656);