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 | |
2136 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)(5,6,8,10)(7,9,11,12)(13,14)(15,18)(16,20)(17,19)(21,22)(23,30)(24,28)(25,29)(26,27)(31,36)(32,35)(33,34);(2,4)(13,15,17,20)(14,18,19,16)(22,27)(23,28)(25,33)(32,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,31)(26,28)(27,30)(29,32)(33,35)(34,36);(5,7,8,11)(6,9,10,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,28)(23,27)(25,32)(26,30)(29,35)(31,34)(33,36);(21,25,26,33)(22,29,27,34)(23,31,28,35)(24,32,30,36)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=gh,e^2=f,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, 605556);