Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| 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 | |
1856 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)(15,16)(18,19)(21,22)(23,28)(24,32)(25,30)(26,27)(29,35)(31,36)(33,34);(6,9)(10,12)(13,14,17,20)(15,18,16,19)(22,27)(23,30)(29,32)(31,33);(5,6)(7,10)(8,9)(11,12)(13,15,17,16)(14,18,20,19)(21,23,26,30)(22,28,27,25)(24,33,35,31)(29,36,32,34);(1,2,3,4)(21,24)(22,29)(23,31)(25,36)(26,35)(27,32)(28,34)(30,33);(5,7,8,11)(6,10,9,12)(13,16,17,15)(14,19,20,18)(21,25,26,28)(22,30,27,23)(24,34,35,36)(29,31,32,33)Polycyclic presentation:
2^9|b^2=g,c^2=fg,d^2=fi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1317371);