Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| 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 | |
1916 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)(6,10)(7,12)(13,14,17,16)(15,18,19,20)(22,26)(23,29)(28,32)(31,35);(15,19)(18,20)(21,22,25,26)(23,27,29,30)(24,32,33,28)(31,36,35,34);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,20)(17,19)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(1,2,3,4)(5,7,9,12)(6,11,10,8)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,36)(33,35);(5,8,9,11)(6,12,10,7)(13,16,17,14)(15,20,19,18)(21,24)(22,28)(23,31)(25,33)(26,32)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=fh,b^2=f,d^2=fi,e^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 628807);