Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
1954 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)(5,6,10,12)(7,9,11,8)(14,17)(15,19)(21,22,26,29)(23,24,30,27)(25,33,34,28)(31,35,36,32);(5,7,10,11)(6,8,12,9)(13,14)(15,18)(16,17)(19,20)(23,30)(24,27)(31,36)(32,35);(5,8,10,9)(6,11,12,7)(13,14)(15,20)(16,17)(18,19)(21,23)(22,27)(24,29)(25,31)(26,30)(28,35)(32,33)(34,36);(1,2,3,4)(5,9,10,8)(6,11,12,7)(21,24,26,27)(22,23,29,30)(25,32,34,35)(28,31,33,36);(5,6)(7,8)(9,11)(10,12)(13,15,16,19)(14,18,17,20)(21,25,26,34)(22,28,29,33)(23,31,30,36)(24,32,27,35)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=fgh,d^2=fghi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2236727);