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 | |
1840 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)(7,9)(8,10)(11,12)(21,22,27,24)(23,30,28,25)(26,32,36,29)(31,33,34,35);(6,7)(8,12)(14,17)(18,20)(21,23,27,28)(22,25,24,30)(26,34,36,31)(29,35,32,33);(6,7)(10,11)(13,14,16,17)(15,18,19,20)(21,24,27,22)(23,25,28,30)(26,32,36,29)(31,33,34,35);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(21,25)(22,28)(23,24)(26,35)(27,30)(29,31)(32,34)(33,36);(5,8)(6,10)(7,11)(9,12)(13,15,16,19)(14,18,17,20)(21,26,27,36)(22,29,24,32)(23,31,28,34)(25,33,30,35)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=h,d^2=g,e^2=h,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, 2987084);