Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #763332 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(13,14)(15,17)(16,19)(18,20)(21,22,25,26)(23,29,31,27)(24,32,34,28)(30,35,33,36);(5,6,8,10)(7,9,11,12)(13,15,18,19)(14,17,20,16)(22,26)(27,29)(28,32)(35,36);(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,23)(22,27)(24,33)(25,31)(26,29)(28,36)(30,34)(32,35);(1,3,4,2)(13,16,18,17)(14,19,20,15)(21,22)(23,27)(24,28)(25,26)(29,31)(30,35)(32,34)(33,36);(5,7,8,11)(6,9,10,12)(13,17,18,16)(14,15,20,19)(21,24)(22,28)(23,30)(25,34)(26,32)(27,35)(29,36)(31,33)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fi,d^2=fg,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 677005);