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 | |
1779 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)(11,21)(12,17)(13,18)(14,19)(15,20)(16,23)(22,35)(24,30)(25,32)(26,34)(27,33)(28,31)(29,36)(37,40)(38,39);(9,11,15,24)(10,16,20,31)(12,22,27,38)(13,23,14,28)(17,29,33,40)(18,30,19,21)(25,36,26,37)(32,39,34,35);(1,3,4,8,5,2,6,7)(9,12)(10,17)(11,22)(13,25)(14,26)(15,27)(16,29)(18,32)(19,34)(20,33)(21,35)(23,36)(24,38)(28,37)(30,39)(31,40);(2,3)(7,8)(9,13,15,14)(10,18,20,19)(11,23,24,28)(12,25,27,26)(16,30,31,21)(17,32,33,34)(22,36,38,37)(29,39,40,35);(9,13,15,14)(10,19,20,18)(11,23,24,28)(12,26,27,25)(16,21,31,30)(17,32,33,34)(22,37,38,36)(29,39,40,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^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 | #72321806, #72321847, #72321918, … |
| maximal normal subgroup of | #72321806, #72321847, #72321918, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252276);