Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480476 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #35149 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2085 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)(5,6,10,16,11,14,8,13)(7,17,19,33,21,34,24,36)(9,15,18,27,23,31,25,32)(12,26,30,22,29,35,28,20);(5,7,18,26,11,21,25,35)(6,12,27,33,14,29,32,36)(8,19,23,20,10,24,9,22)(13,28,31,34,16,30,15,17);(2,4)(5,8,11,10)(6,13,14,16)(7,19,21,24)(9,18,23,25)(12,28,29,30)(15,27,31,32)(17,33,34,36)(20,35,22,26);(6,14)(12,29)(13,16)(15,31)(17,34)(27,32)(28,30)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286439, #72286501, #72286828, … |
| maximal normal subgroup of | #72286439, #72286501, #72286828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401928);