Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219532 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,12,9,24)(7,16,19,26,20,28,17,13)(8,21,22,27,23,30,25,14)(18,29,32,35,33,36,34,31);(1,2)(3,4)(6,12)(7,17)(9,10)(13,28)(16,26)(18,32)(19,20)(21,30)(22,25)(29,31)(33,34)(35,36);(2,4)(5,7,9,17,11,20,10,19)(6,13,15,16,12,26,24,28)(8,18,22,32,23,33,25,34)(14,29,30,31,27,36,21,35);(5,8)(6,14,12,27)(7,18)(9,22)(10,25)(11,23)(13,29,26,36)(15,30,24,21)(16,31,28,35)(17,32)(19,34)(20,33)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291536);