Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498807 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1542 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)(7,18)(8,12)(9,17)(10,16)(11,14)(13,21)(15,23)(19,30)(20,31)(22,32)(24,28)(25,27)(26,36)(29,35)(33,34);(6,12)(7,19)(8,23)(9,10)(13,29)(14,27)(15,16)(17,28)(18,33)(20,35)(21,22)(24,25)(26,32)(30,31);(2,4)(5,7)(6,13)(8,20)(9,19)(10,21)(11,22)(12,26)(14,30)(15,31)(16,32)(17,18)(23,34)(24,36)(25,35)(27,33)(28,29);(6,14)(12,27)(13,30)(15,28)(16,17)(18,32)(26,33)(29,31)Polycyclic presentation:
2^9|e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239802, #72239814, #72239838, … |
| maximal normal subgroup of | #72239802, #72239814, #72239838, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406003);