Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #917169 |
| automorphism | ? |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2270 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)(13,14)(15,18)(16,17)(19,20)(21,22)(23,29)(24,31)(25,33)(26,27)(28,34)(30,32)(35,36);(7,11)(10,12)(13,15,17,20)(14,18,16,19)(22,27)(24,29)(28,33)(32,35);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,18)(17,20)(21,23,26,31)(22,24,27,29)(25,30,34,36)(28,32,33,35);(1,2,3,4)(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,18,20,19)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,36)(34,35);(21,25,26,34)(22,28,27,33)(23,30,31,36)(24,32,29,35)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 853971);