Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
841 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,12,15,25,35,24,34,11,17,10,16,8,14,9,26)(7,18,20,33,19,30,27,13,23,32,21,29,22,36,28,31);(1,2,3,4)(5,7,11,23)(6,13,17,31)(8,19,25,22)(9,20,24,21)(10,27,12,28)(14,29,35,33)(15,30,16,36)(18,26,32,34);(5,8,11,25)(6,14,17,35)(7,19,23,22)(9,12,24,10)(13,29,31,33)(15,34,16,26)(18,36,32,30)(20,28,21,27);(5,9,25,12,11,24,8,10)(6,15,14,26,17,16,35,34)(7,20,22,28,23,21,19,27)(13,30,29,32,31,36,33,18)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=i,d^2=fh,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,h^b=hi,d^c=di,e^c=ei,g^c=gi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
| maximal quotient of | #72282356 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402739);