Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2322 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)(6,9)(7,11)(13,14,17,20)(15,19,16,18)(21,22)(23,30)(24,32)(25,28)(26,29)(27,35)(31,36)(33,34);(7,11)(10,12)(15,16)(18,19)(21,23,26,28)(22,25,29,30)(24,33,35,31)(27,34,32,36);(5,6)(7,10)(8,9)(11,12)(13,15,17,16)(14,18,20,19)(21,22)(23,25)(24,27)(26,29)(28,30)(31,36)(32,35)(33,34);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,24,26,35)(22,27,29,32)(23,31,28,33)(25,36,30,34);(5,7,8,11)(6,10,9,12)(13,16,17,15)(14,19,20,18)(21,25)(22,28)(23,29)(24,34)(26,30)(27,31)(32,33)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=h,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3945204);