Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,26,30,25,28,34,36)(23,27,32,35,31,24,29,33);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23)(22,27)(24,28)(25,31)(26,32)(29,34)(30,35)(33,36);(2,4)(22,28)(23,31)(29,32)(30,36);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(23,31)(24,27)(29,32)(33,35);(6,12)(10,13)(11,20)(18,19)(21,24,26,33,25,27,34,35)(22,29,30,23,28,32,36,31)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=dfi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6307170);