Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563337 |
| 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 | |
2693 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,19)(8,25)(9,14)(10,17)(11,16)(12,18)(13,22)(15,28)(20,29)(21,30)(23,32)(24,31)(26,35)(27,34)(33,36);(1,2)(3,4)(5,7,10,22,12,24,11,23)(6,13,16,19,18,32,17,31)(8,26,9,20,28,36,27,21)(14,33,25,29,34,35,15,30);(5,8,10,9,12,28,11,27)(6,14,16,25,18,34,17,15)(7,20,23,26,24,21,22,36)(13,29,31,33,32,30,19,35);(2,4)(5,8,11,27,12,28,10,9)(6,14,17,15,18,34,16,25)(7,21,22,26,24,20,23,36)(13,30,19,33,32,29,31,35);(5,9,10,28,12,27,11,8)(6,15,16,14,18,25,17,34)(7,20,22,36,24,21,23,26)(13,30,19,33,32,29,31,35)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=gi,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470535);