Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #114839 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | #60750 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1323 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4,7,8)(5,6)(9,10,14,18)(11,19)(12,15,13,17)(16,22)(20,24)(21,23);(1,3)(2,4)(5,7)(6,8)(10,15)(11,20)(12,13)(17,18)(21,22)(23,24);(1,4)(2,3)(5,8)(6,7)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,24)(18,19)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,d^c=dh,f^c=fi,g^c=gh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72214321, #72214400, #72214512, … |
| maximal normal subgroup of | #72214321, #72214400, #72214512, … |
| maximal quotient of | C2×#114839, #72311484, #72311554, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22035);