Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| 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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,10,12)(7,16,20,13)(8,21,23,30)(9,11,25,15)(14,24,27,22)(17,26,19,29)(18,33,34,36)(28,32,31,35);(1,2)(3,4)(6,11)(7,17)(9,25)(12,15)(14,21)(18,32)(19,20)(22,24)(26,29)(27,30)(33,36)(34,35);(2,4)(6,12)(11,15)(13,16)(14,27)(21,30)(26,29)(28,31)(33,36);(2,4)(5,7,10,20)(6,13,12,16)(8,18,23,34)(9,19,25,17)(11,26,15,29)(14,28,27,31)(21,36,30,33)(22,32,24,35);(5,8)(6,14)(7,18)(9,24)(10,23)(11,21)(12,27)(13,28)(15,30)(16,31)(17,32)(19,35)(20,34)(22,25)(26,36)(29,33)Polycyclic presentation:
2^9|a^2=i,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,h^b=hi,h^d=hi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421375);