Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,11,16,12,18,10,17)(7,19,22,14,8,23,21,13)(9,25,27,32,28,31,26,15)(20,30,35,36,24,29,34,33);(1,3,4,2)(5,7,10,21,12,8,11,22)(6,13,16,19,18,14,17,23)(9,20,26,34,28,24,27,35)(15,29,25,33,32,30,31,36);(2,3)(5,8)(6,14,18,13)(7,12)(9,24)(10,22)(11,21)(15,30,32,29)(16,23,17,19)(20,28)(25,36,31,33)(26,35)(27,34);(5,9,10,26,12,28,11,27)(6,15,16,25,18,32,17,31)(7,20,21,34,8,24,22,35)(13,29,19,33,14,30,23,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fh,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281947);