Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(6,9)(10,12)(14,17)(18,19)(23,29)(24,32)(26,33)(28,34)(30,35)(31,36);(5,6,8,9)(7,10,11,12)(13,14,16,17)(15,19,20,18)(21,22)(23,30)(24,28)(25,27)(26,31)(29,35)(32,34)(33,36);(7,11)(10,12)(13,15)(14,18)(16,20)(17,19)(21,23,24,29,25,31,32,36)(22,26,28,33,27,30,34,35);(5,7,8,11)(6,10,9,12)(15,20)(18,19)(22,27)(26,30)(28,34)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,d^2=egi,f^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=di,d^c=de80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288605);