Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| 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 | |
780 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)(5,6,11,17)(7,18)(8,23)(9,15,12,29)(10,25,27,30)(13,26)(14,33)(16,19,34,28)(20,36)(21,32)(22,35)(24,31);(5,7,19,24,9,20,27,21,11,22,28,33,12,8,10,26)(6,13,30,36,29,14,34,18,17,32,25,23,15,31,16,35);(2,4)(5,8,10,24,12,22,28,26,11,20,27,33,9,7,19,21)(6,14,16,23,15,13,25,18,17,31,34,36,29,32,30,35);(5,9,11,12)(6,15,17,29)(7,20,22,8)(10,19,27,28)(13,31,32,14)(16,25,34,30)(18,36,35,23)(21,33,26,24)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=e,d^2=i,e^2=h,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249916, #72249929, #72249966, … |
| maximal normal subgroup of | #72249916, #72249929, #72249966, … |
| maximal quotient of | #72286002, #72309634, #72310175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402119);