Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419800 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1208 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)(5,6,9,15,11,17,10,16)(7,18,20,29,22,32,21,12)(8,14,24,13,26,30,25,31)(19,27,23,35,33,36,34,28);(5,7,10,20,11,22,9,21)(6,12,16,18,17,29,15,32)(8,23,25,33,26,34,24,19)(13,28,14,27,31,35,30,36);(5,8)(6,13)(7,19,22,33)(9,25)(10,24)(11,26)(12,27,29,36)(14,15)(16,30)(17,31)(18,28,32,35)(20,34,21,23);(2,4)(5,8)(6,14)(7,19,22,33)(9,24)(10,25)(11,26)(12,28,29,35)(13,15)(16,31)(17,30)(18,27,32,36)(20,23,21,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ef,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339192);