Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4026 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)(7,17)(8,22,25,30)(9,16)(10,15)(11,14)(12,19)(13,24,29,23)(18,26,33,36)(20,27)(21,28)(31,32,35,34);(5,7,10,19,11,21,9,20)(6,12,15,28,14,27,16,17)(8,18,24,32,25,33,23,34)(13,26,22,35,29,36,30,31);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,29)(15,22)(16,30)(17,31)(19,32)(20,34)(21,33)(27,36)(28,35);(6,14)(12,27)(13,29)(15,16)(17,28)(22,30)(26,36)(31,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,e^b=ei,f^c=fi,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72237181, #72237184, #72237188, … |
| maximal normal subgroup of | #72237181, #72237184, #72237188, … |
| maximal quotient of | #72239101, #72257492, #72257591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260652);