Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#481 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1070 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)(7,16,19,32)(8,13)(9,21)(10,25)(11,15)(12,18,28,34)(14,24)(17,33,26,30)(20,31)(22,29,36,27)(23,35);(5,7,10,26,11,19,8,17)(6,12,13,27,15,28,25,29)(9,22,24,18,23,36,20,34)(14,30,21,16,31,33,35,32);(2,4)(9,23)(14,31)(16,32)(18,34)(20,24)(21,35)(22,36)(30,33);(5,8,11,10)(6,13,15,25)(7,17,19,26)(9,20,23,24)(12,27,28,29)(14,21,31,35)(16,33,32,30)(18,22,34,36)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=i,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,f^b=fi,g^b=gi,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401465);