Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×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 | D4×C22⋊C4 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1947 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16)(7,17)(8,21,26,33)(9,13)(10,14,29,15)(12,24,32,25)(18,23)(19,27)(20,28)(22,30)(31,36)(34,35);(1,2)(3,4)(5,7)(6,9)(8,22)(10,13)(11,20)(12,30)(14,28)(15,17)(16,27)(18,26)(19,29)(21,34)(23,32)(24,35)(25,31)(33,36);(2,4)(5,8)(6,12)(7,18)(9,23)(10,24)(11,26)(13,31)(14,21)(15,33)(16,32)(17,34)(19,35)(20,22)(25,29)(27,30)(28,36);(5,9)(6,13)(7,15)(8,23)(10,28)(11,27)(12,31)(14,20)(16,19)(17,29)(18,33)(21,22)(24,36)(25,34)(26,30)(32,35)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,e^d=ei,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290330);