Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563346 |
| 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 | |
2519 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)(5,6,11,17)(7,18,22,33)(8,13,24,14)(9,25,26,32)(10,16,27,15)(12,21,30,20)(19,31,35,29)(23,36,34,28);(2,3)(5,7,10,20,11,22,27,21)(6,12,15,18,17,30,16,33)(8,23,9,19,24,34,26,35)(13,31,32,36,14,29,25,28);(1,2,4,3);(1,3,4,2)(5,8,10,9,11,24,27,26)(6,13,16,25,17,14,15,32)(7,19,21,23,22,35,20,34)(12,28,18,31,30,36,33,29);(5,9,10,24,11,26,27,8)(6,14,15,25,17,13,16,32)(7,19,20,34,22,35,21,23)(12,29,18,28,30,31,33,36)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=h,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,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521299);