Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
667 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,15,12,16,10,18)(7,9,21,29,20,30,19,13)(8,22,26,34,27,14,25,33)(17,28,23,31,36,32,35,24);(5,7,10,19,12,20,11,21)(6,13,15,9,16,29,18,30)(8,23,25,17,27,35,26,36)(14,32,33,24,22,28,34,31);(5,8,12,27)(6,14,16,22)(7,17,20,36)(9,28,30,32)(10,26,11,25)(13,31,29,24)(15,34,18,33)(19,23,21,35);(2,4)(5,9,10,29,12,30,11,13)(6,7,18,21,16,20,15,19)(8,24,25,28,27,31,26,32)(14,23,34,36,22,35,33,17)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=i,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386366);