Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#577 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,10,16,11,15,9,24)(7,17,20,12,21,29,25,28)(8,14,22,30,23,31,26,13)(18,34,19,33,36,27,35,32);(5,7,11,21)(6,12,15,28)(8,18,23,36)(9,25,10,20)(13,27,30,34)(14,32,31,33)(16,29,24,17)(19,26,35,22);(2,4)(5,8)(6,13)(7,18)(9,22)(10,26)(11,23)(12,27)(14,24)(15,30)(16,31)(17,32)(19,25)(20,35)(21,36)(28,34)(29,33);(5,8)(6,14)(7,19)(9,26)(10,22)(11,23)(12,27)(13,24)(15,31)(16,30)(17,33)(18,25)(20,36)(21,35)(28,34)(29,32);(6,15)(12,28)(13,30)(14,31)(16,24)(17,29)(27,34)(32,33)Polycyclic presentation:
2^9|a^2=h,b^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,d^b=df,f^b=fi,g^c=gi,h^c=hi,f^d=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72317468 |
| maximal normal subgroup of | #72317468 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286403);