Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| 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 | #16500 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2373 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)(7,9)(8,10)(11,12)(13,14)(15,19)(16,17)(18,20)(21,22)(23,27)(24,26)(25,28)(29,35)(30,31)(32,34)(33,36);(8,12)(10,11)(13,15,18,16)(14,19,20,17)(22,24)(27,31)(28,32)(35,36);(6,7)(10,11)(14,20)(15,16)(21,23)(22,27)(24,31)(25,33)(26,30)(28,36)(29,34)(32,35);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,18,15)(14,19,20,17)(21,24,26,22)(23,27,30,31)(25,32,34,28)(29,35,33,36);(5,8)(6,10)(7,11)(9,12)(13,17,18,19)(14,15,20,16)(21,25,26,34)(22,28,24,32)(23,29,30,33)(27,35,31,36)Polycyclic presentation:
2^9|b^2=h,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4990275);