Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519063 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(5,6)(7,18)(8,13)(9,24)(10,16)(11,17)(12,20)(14,25)(15,22)(19,33)(21,28)(23,27)(26,30)(29,35)(31,36)(32,34);(2,3)(5,7,10,20,11,8,22,21)(6,12,15,18,17,28,16,13)(9,19,25,34,26,23,36,35)(14,27,24,32,31,33,30,29);(1,2,4,3);(1,3,4,2)(5,8)(6,13)(7,11)(9,23)(10,21)(12,15)(14,29)(16,28)(17,18)(19,26)(20,22)(24,27)(25,35)(30,33)(31,32)(34,36);(5,9,10,25,11,26,22,36)(6,14,15,24,17,31,16,30)(7,19,20,34,8,23,21,35)(12,27,18,32,28,33,13,29)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=g,e^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470485);