Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,14,11,15,9,25)(7,16,18,12,19,29,26,28)(8,20,24,34,22,13,23,31)(17,30,21,27,32,33,35,36);(1,2)(3,4)(5,7)(6,12)(8,21)(9,26)(10,18)(11,19)(13,30)(14,29)(15,28)(16,25)(17,23)(20,33)(22,35)(24,32)(27,31)(34,36);(2,4)(5,8,9,23,11,22,10,24)(6,13,14,31,15,20,25,34)(7,17,18,21,19,32,26,35)(12,27,16,30,28,36,29,33);(8,22)(13,20)(17,32)(21,35)(23,24)(27,36)(30,33)(31,34)Polycyclic presentation:
2^9|a^2=h,c^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=di,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72316897, #72316903, #72316909, … |
| maximal normal subgroup of | #72316897, #72316903, #72316909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265163);