Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476293 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,16,11,15,10,14)(7,17,20,32,21,12,19,28)(8,22,24,31,26,30,25,13)(18,34,35,27,36,29,23,33);(1,2)(3,4)(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,29)(14,28)(15,17)(16,32)(18,25)(22,27)(24,36)(26,35)(30,33)(31,34);(2,4)(5,8)(6,13)(7,18)(9,25)(10,24)(11,26)(12,27)(14,22)(15,31)(16,30)(17,33)(19,35)(20,23)(21,36)(28,34)(29,32);(6,14)(7,19)(9,10)(12,17)(13,30)(15,16)(18,35)(20,21)(22,31)(23,36)(24,25)(29,34)Polycyclic presentation:
2^9|a^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,f^d=fi,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, 383489);