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 | |
1984 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,7)(8,12)(15,19)(18,20)(21,22,27,30)(23,25,24,28)(26,29,35,36)(31,33,32,34);(2,4)(6,7)(10,11)(14,17)(18,20)(21,23,27,24)(22,28,30,25)(26,32,35,31)(29,33,36,34);(1,2,3,4)(5,6,9,7)(8,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,24,27,23)(22,25,30,28)(26,32,35,31)(29,33,36,34);(5,7,9,6)(8,11,12,10)(13,15)(14,18)(16,19)(17,20)(21,25)(22,24)(23,30)(26,34)(27,28)(29,31)(32,36)(33,35);(5,8)(6,10)(7,11)(9,12)(21,26,27,35)(22,29,30,36)(23,31,24,32)(25,33,28,34)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=ghi,d^2=fg,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, 3014020);