Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
2010 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)(15,19)(18,20)(21,22,26,29)(23,27,24,30)(25,33,34,28)(31,36,32,35);(5,7)(6,10)(8,11)(9,12)(14,17)(18,20)(21,23,26,24)(22,27,29,30)(25,31,34,32)(28,35,33,36);(5,7,9,12)(6,11,8,10)(13,14)(15,18)(16,17)(19,20)(21,24,26,23)(22,27,29,30)(25,32,34,31)(28,35,33,36);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,18,17,20)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,35)(33,34);(5,8,9,6)(7,10,12,11)(21,25)(22,28)(23,31)(24,32)(26,34)(27,35)(29,33)(30,36)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fg,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2980709);