Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #189674 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#380 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#407 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1640 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)(7,17)(8,20,23,13)(9,12)(10,25)(11,16)(14,19)(15,26)(18,29,32,31)(21,33,35,27)(22,30,36,34)(24,28);(1,2)(3,4)(5,7,10,9,11,19,26,24)(6,12,15,17,16,28,25,14)(8,18,22,21,23,32,36,35)(13,27,30,31,20,33,34,29);(2,4)(5,8)(6,13)(7,18)(9,21)(10,22)(11,23)(12,27)(14,29)(15,30)(16,20)(17,31)(19,32)(24,35)(25,34)(26,36)(28,33);(5,9,10,19,11,24,26,7)(6,14,15,12,16,17,25,28)(8,21,22,32,23,35,36,18)(13,29,30,27,20,31,34,33)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267389);