Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499263 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13316 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
836 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)(7,12)(8,18)(9,22)(10,26)(11,15)(13,21)(14,23)(16,33)(17,32)(19,31)(20,35)(24,27)(25,30)(28,34)(29,36);(1,2)(3,4)(5,7,11,10)(6,12,15,26)(8,16,20,19)(9,23,25,17)(13,27,28,29)(14,30,32,22)(18,31,35,33)(21,36,34,24);(2,4)(5,8)(6,13)(7,16)(9,24)(10,19)(11,20)(12,27)(14,31)(15,28)(17,34)(18,22)(21,23)(25,36)(26,29)(30,35)(32,33);(8,19,20,16)(9,25)(13,27,28,29)(14,32)(17,23)(18,31,35,33)(21,24,34,36)(22,30)Polycyclic presentation:
2^9|b^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,f^b=fi,d^c=dh,e^c=ei,f^c=fh,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72231495, #72250066, #72252788, … |
| maximal normal subgroup of | #72231495, #72250066, #72252788, … |
| maximal quotient of | #72250666, C2×#499265, #72282356 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406461);