Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #381 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10,13,19,14,17,12,18)(11,20,21,24,22,15,23,16);(1,2)(3,4)(5,6,7,8)(9,11)(10,15)(12,21)(13,23)(14,22)(16,18)(17,20)(19,24);(2,4)(9,11)(10,16,17,24)(12,21)(13,23)(14,22)(15,18,20,19)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,g^b=gi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72235489, #72235493, #72235504, … |
| maximal normal subgroup of | #72235489, #72235493, #72235504, … |
| maximal quotient of | #72235541, C2×#123696, #72235633 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30892);