Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#442 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,8,14,11,16,9,12)(7,17,20,31,21,34,18,33)(10,24,22,15,25,30,23,32)(13,26,27,19,29,36,28,35);(1,2,4,3)(6,12,16,14)(10,25)(13,27,29,28)(15,30,32,24)(17,33,34,31)(19,35)(22,23)(26,36);(1,3,4,2)(5,7)(6,13)(8,18)(9,20)(10,26)(11,21)(12,27)(14,28)(15,31)(16,29)(17,30)(19,23)(22,35)(24,34)(25,36)(32,33);(5,8,11,9)(6,14,16,12)(7,18,21,20)(10,22,25,23)(13,28,29,27)(15,30,32,24)(17,33,34,31)(19,26,35,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=eh,d^2=i,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,f^b=fi,e^c=ei,f^c=fhi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239532, #72239548, #72239554, … |
| maximal normal subgroup of | #72239532, #72239548, #72239554, … |
| maximal quotient of | #72285467, #72285490, #72310157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402087);