Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #392 |
| prime radical | C24 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(6,9)(7,11)(10,12)(18,25)(21,27);(5,6,7,9,8,10,11,12)(13,14)(15,21,22,27)(16,20)(17,19)(18,23,25,28)(24,26);(13,15)(14,18,19,25)(16,23)(17,22)(20,21,26,27)(24,28)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#121335, #72211654, #72214494, … |
| maximal normal subgroup of | C2×#121335, #72211654, #72214494, … |
| maximal quotient of | #72211228, C2×#121335, #72214224, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28531);