Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
958 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)(5,6,12,16,11,17,10,18)(7,19,24,32,23,34,22,13)(8,25,27,36,21,14,26,35)(9,28,20,33,30,31,29,15);(5,7,11,23)(6,13)(8,9,21,30)(10,22,12,24)(14,33)(15,25)(16,19)(17,32)(18,34)(20,26,29,27)(28,36)(31,35);(2,4)(5,8)(6,14)(7,20)(9,22)(10,26)(11,21)(12,27)(13,31)(15,19)(16,35)(17,25)(18,36)(23,29)(24,30)(28,32)(33,34);(5,9,10,29,11,30,12,20)(6,15,16,28,17,33,18,31)(7,21,22,27,23,8,24,26)(13,25,19,36,32,14,34,35)Polycyclic presentation:
2^9|a^2=h,b^2=efh,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272516);