Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2920 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10)(11,20)(12,17)(13,18)(14,27)(15,19)(16,22)(21,34)(23,30)(24,32)(25,33)(26,37)(28,31)(29,35)(36,39)(38,40);(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,29)(18,30)(19,31)(20,27)(24,35)(25,36)(26,38)(32,39)(33,40)(34,37);(1,3,4,8,5,2,6,7)(9,12)(10,17)(11,21)(13,24)(14,26)(15,25)(16,29)(18,32)(19,33)(20,34)(22,35)(23,36)(27,37)(28,38)(30,39)(31,40);(2,3)(7,8)(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,24,25,26)(16,30,31,20)(17,32,33,37)(21,35,36,38)(29,39,40,34);(10,19)(12,25)(16,31)(18,27)(20,30)(21,36)(24,26)(35,38)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=h,f^2=h,g^2=i,b^a=bf,d^a=di,e^a=eh,f^a=fh,f^b=fh,d^c=di,e^c=eh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72320825, #72320879, #72321035, … |
| maximal normal subgroup of | #72320825, #72320879, #72321035, … |
| maximal quotient of | #72343800 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252270);