Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #112861 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | #35050 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,17,19,14)(12,15)(13,16)(18,22,24,21)(20,23);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23);(3,7)(5,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=dh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72231799, #72236977, #72236993, … |
| maximal normal subgroup of | #72231799, #72236977, #72236993, … |
| maximal quotient of | #72258089 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 20057);