Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1900 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)(7,12)(8,9)(10,11)(15,19)(18,20)(21,22,26,29)(23,25,31,27)(24,32,35,28)(30,34,33,36);(6,11)(9,12)(14,17)(18,20)(21,23,26,31)(22,27,29,25)(24,33,35,30)(28,36,32,34);(5,7,10,8)(6,12,11,9)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,30)(25,29)(26,31)(28,34)(32,36)(33,35);(1,2,3,4)(5,8,10,7)(6,12,11,9)(13,15)(14,18)(16,19)(17,20)(21,24)(22,28)(23,30)(25,36)(26,35)(27,34)(29,32)(31,33);(5,9,10,12)(6,8,11,7)(21,25,26,27)(22,23,29,31)(24,34,35,36)(28,33,32,30)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fg,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2831967);