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) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14885 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2835 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)(7,11)(15,19)(17,20)(21,22)(23,29)(24,26)(25,28)(27,30)(31,36)(32,34)(33,35);(6,9)(10,12)(13,14,16,18)(15,17,19,20)(22,24)(23,30)(28,32)(31,33);(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,27)(24,29)(25,33)(26,30)(28,36)(31,34)(32,35);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(21,24,26,22)(23,29,30,27)(25,32,34,28)(31,36,33,35);(5,7)(6,10)(8,11)(9,12)(13,15)(14,17)(16,19)(18,20)(21,25,26,34)(22,28,24,32)(23,31,30,33)(27,35,29,36)Polycyclic presentation:
2^9|b^2=gh,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, 4505630);