Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500891 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,13)(7,19,22,36)(8,23)(9,12)(10,25)(14,29,20,33)(15,24)(17,18)(21,26,27,35)(28,31,34,30);(1,2,3,4)(5,6,12,18,11,17,9,13)(7,20,27,28,22,14,21,34)(8,24,10,16,25,15,23,32)(19,30,26,29,36,31,35,33);(2,4)(5,7)(6,14)(8,19)(9,21)(10,26)(11,22)(12,27)(13,28)(15,30)(16,33)(17,20)(18,34)(23,35)(24,31)(25,36)(29,32);(5,8,9,23,11,25,12,10)(6,15,13,16,17,24,18,32)(7,19,21,35,22,36,27,26)(14,30,28,33,20,31,34,29)Polycyclic presentation:
2^9|a^2=eh,b^2=eg,d^2=e,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fhi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72238779, #72239803, #72239829, … |
| maximal normal subgroup of | #72238779, #72239803, #72239829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413661);