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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2026 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)(5,6,9,8)(7,10,11,12)(13,14)(15,16)(17,19)(18,20)(21,22)(23,28)(24,25)(26,27)(29,34)(30,36)(31,33)(32,35);(7,11)(10,12)(13,15,17,20)(14,16,19,18)(22,27)(25,29)(28,35)(31,36);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,19)(17,20)(21,23)(22,28)(24,33)(25,31)(26,32)(27,35)(29,36)(30,34);(1,2,3,4)(21,24)(22,29)(23,30)(25,27)(26,34)(28,31)(32,33)(35,36);(5,8,9,6)(7,12,11,10)(13,16,17,18)(14,15,19,20)(21,25,26,29)(22,24,27,34)(23,31,32,36)(28,33,35,30)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=fi,e^2=fh,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, 628728);