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 | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,9,16,11,13)(7,12,17,27,19,28,20,31)(8,21,24,32,23,14,25,29)(18,26,33,30,34,36,22,35);(1,2)(3,4)(5,7)(6,12)(8,22)(9,19)(10,20)(11,17)(13,27)(14,30)(15,31)(16,28)(18,25)(21,35)(23,33)(24,34)(26,29)(32,36);(2,4)(6,13)(7,17)(8,23)(10,11)(14,29)(15,16)(18,33)(19,20)(21,32)(22,34)(27,31)(30,35);(5,8,9,23)(6,14,16,21)(7,18,19,34)(10,25,11,24)(12,26,28,36)(13,29,15,32)(17,22,20,33)(27,35,31,30)Polycyclic presentation:
2^9|a^2=f,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, 383494);