Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,18,12,16,10,17)(7,19,22,32,23,13,29,31)(8,24,26,14,21,34,25,33)(9,15,27,36,20,35,28,30);(1,2)(3,4)(5,7)(6,13)(8,20)(9,21)(10,22)(11,29)(12,23)(14,30)(15,24)(16,19)(17,31)(18,32)(25,28)(26,27)(33,36)(34,35);(2,4)(5,8,12,21)(6,14,16,33)(7,20,23,9)(10,26,11,25)(13,30,19,36)(15,31,35,32)(17,34,18,24)(22,27,29,28);(5,9,12,20)(6,15,16,35)(7,21,23,8)(10,28,11,27)(13,24,19,34)(14,32,33,31)(17,30,18,36)(22,25,29,26)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,h^b=hi,d^c=de,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287016);