Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9938 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1438 |
| prime radical | #34957 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4,7,8)(5,6)(9,10)(11,17)(12,20)(13,16)(14,18)(15,22)(19,24)(21,23);(1,3)(2,4)(5,7)(6,8)(9,11)(10,14)(12,21)(13,19)(15,24)(16,23)(17,20)(18,22);(1,4)(2,3)(5,8)(6,7)(14,23)(17,24)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dhi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231222, #72232727, #72234080, … |
| maximal normal subgroup of | #72231222, #72232727, #72234080, … |
| maximal quotient of | #72231150, #72231342, #72233586, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 19777);