Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C42×D4 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10)(11,20)(12,24)(13,28)(14,19)(15,18)(16,22)(17,25)(21,35)(23,32)(26,33)(27,39)(29,31)(30,40)(34,36)(37,38);(9,11,14,22,15,23,13,29)(10,16,19,32,18,31,28,20)(12,21,27,37,25,36,26,40)(17,30,33,35,24,38,39,34);(9,12,14,27,15,25,13,26)(10,17,19,33,18,24,28,39)(11,21,22,37,23,36,29,40)(16,30,32,35,31,38,20,34);(1,3,4,8,5,2,6,7)(10,18)(12,25)(16,31)(19,28)(20,32)(21,36)(26,27)(37,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=h,f^2=i,g^2=h,b^a=be,c^a=ch,d^a=dh,e^a=eh,e^b=eh,d^c=dh176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91210);