Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395174 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1119 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)(9,16)(10,14)(11,15)(12,19)(13,25)(18,29)(20,28)(21,27)(23,30)(24,31)(26,33)(32,34)(35,36);(2,4)(5,7,10,20,11,21,9,19)(6,12,16,27,15,28,14,17)(8,18,25,35,23,33,24,34)(13,29,22,32,31,26,30,36);(5,8)(6,13)(7,18)(9,24)(10,25)(11,23)(12,26)(14,30)(15,31)(16,22)(17,32)(19,34)(20,35)(21,33)(27,36)(28,29);(6,14,15,16)(8,23)(12,17,28,27)(13,22,31,30)(18,33)(24,25)(26,36,29,32)(34,35)Polycyclic presentation:
2^9|b^2=h,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231526, #72252980, #72256881, … |
| maximal normal subgroup of | #72231526, #72252980, #72256881, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318512);