Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| 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 | |
2652 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)(14,26)(15,22)(19,33)(21,28)(23,27)(25,31)(29,35)(30,36)(32,34);(2,3)(5,7,10,20,11,8,22,21)(6,12,15,18,17,28,16,13)(9,19,26,34,25,23,36,35)(14,27,24,32,30,33,31,29);(1,2,4,3)(9,25)(14,30)(19,23)(24,31)(26,36)(27,33)(29,32)(34,35);(1,3,4,2)(5,8)(6,13)(7,11)(9,23)(10,21)(12,15)(14,29)(16,28)(17,18)(19,25)(20,22)(24,27)(26,35)(30,32)(31,33)(34,36);(5,9,22,36,11,25,10,26)(6,14,16,31,17,30,15,24)(7,19,21,35,8,23,20,34)(12,27,13,29,28,33,18,32)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=g,e^2=fi,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,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470505);