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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2263 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)(7,11)(8,9)(10,12)(13,14)(15,18)(16,20)(17,19)(21,22)(23,24)(25,29)(26,27)(28,31)(30,32)(33,36)(34,35);(5,6)(7,11)(8,9)(10,12)(13,15,17,20)(14,16,19,18)(22,27)(24,31)(29,35)(32,36);(6,9)(10,11)(15,20)(16,18)(21,23)(22,24)(25,33)(26,28)(27,31)(29,36)(30,34)(32,35);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,20)(21,24)(22,28)(23,27)(25,32)(26,31)(29,33)(30,35)(34,36);(5,7)(6,10)(8,12)(9,11)(13,16,17,18)(14,15,19,20)(21,25)(22,29)(23,30)(24,32)(26,34)(27,35)(28,33)(31,36)Polycyclic presentation:
2^9|b^2=gh,d^2=fghi,e^2=gh,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, 4992125);