Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419919 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| 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 | |
973 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)(5,6,10,28,12,16,11,15)(7,17,20,30,19,13,9,27)(8,21,26,31,24,32,25,14)(18,34,35,29,22,36,23,33);(5,7,10,20,12,19,11,9)(6,13,15,30,16,17,28,27)(8,22,25,35,24,18,26,23)(14,29,21,36,31,33,32,34);(5,8)(6,14)(7,18)(9,23)(10,26)(11,25)(12,24)(13,29)(15,32)(16,31)(17,33)(19,22)(20,35)(21,28)(27,36)(30,34);(2,4)(5,9,10,19,12,20,11,7)(6,13,15,27,16,17,28,30)(8,23,26,22,24,35,25,18)(14,29,32,36,31,33,21,34)Polycyclic presentation:
2^9|a^2=efg,b^2=f,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344189);