Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,8,9)(7,11,12,10)(13,14,16,18)(15,19,20,17)(21,22)(23,27)(24,26)(25,32)(28,31)(29,34)(30,36)(33,35);(6,9)(10,11)(13,15)(14,17)(16,20)(18,19)(21,23,26,31)(22,27,24,28)(25,33,34,30)(29,36,32,35);(6,9)(7,12)(13,15,16,20)(14,17,18,19)(21,23)(22,28)(24,27)(25,30)(26,31)(29,36)(32,35)(33,34);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,15)(14,17)(16,20)(18,19)(21,24,26,22)(23,28,31,27)(25,32,34,29)(30,36,33,35);(5,6)(7,11)(8,9)(10,12)(13,14)(15,19)(16,18)(17,20)(21,25)(22,29)(23,30)(24,32)(26,34)(27,35)(28,36)(31,33)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=fgh,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4049416);