Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C2×D42 |
| prime radical | C2×#3147 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6,11,17,12,18,10,16)(7,19,22,32,23,31,28,13)(8,24,9,15,26,33,25,14)(20,29,35,30,21,34,27,36);(1,3,4,2)(5,7,12,23)(6,13,18,32)(8,20,26,21)(9,27,25,35)(10,22,11,28)(14,29,15,34)(16,19,17,31)(24,36,33,30);(5,8,10,25,12,26,11,9)(6,14,17,24,18,15,16,33)(7,20,22,35,23,21,28,27)(13,29,31,36,32,34,19,30);(5,9,11,26,12,25,10,8)(6,15,17,33,18,14,16,24)(7,21,22,27,23,20,28,35)(13,30,19,34,32,36,31,29);(5,10,12,11)(6,16,18,17)(7,22,23,28)(8,25,26,9)(13,19,32,31)(14,33,15,24)(20,35,21,27)(29,30,34,36)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,f^b=fi,h^b=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72224796, #72224802, #72224808, … |
| maximal normal subgroup of | #72224796, #72224802, #72224808, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6290004);