Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476297 |
| 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 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,17,12,16,11,28)(7,18,20,33,22,13,21,31)(8,15,24,32,9,26,25,14)(19,35,27,29,36,30,23,34);(1,2)(3,4)(5,7)(6,13)(8,23)(9,27)(10,21)(11,20)(12,22)(14,30)(15,29)(16,18)(17,33)(19,25)(24,36)(26,34)(28,31)(32,35);(2,4)(5,8,12,9)(6,14,16,32)(7,19,22,36)(10,25,11,24)(13,29,18,34)(15,17,26,28)(20,23,21,27)(30,31,35,33);(5,9,12,8)(6,15,16,26)(7,19,22,36)(10,24,11,25)(13,30,18,35)(14,28,32,17)(20,23,21,27)(29,33,34,31)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,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=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383496);