Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #386 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24);(2,4)(9,11,12,19,13,18,21,24)(10,14,16,22,15,17,20,23);(9,10)(11,14,18,17)(12,20)(13,15)(16,21)(19,23,24,22)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,h^2=i,b^a=bd,c^a=ce,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,f^b=fi,f^c=fi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72280523, #72280639, #72302309, … |
| maximal normal subgroup of | #72280523, #72280639, #72302309, … |
| maximal quotient of | #72280976, #72303279, #72303588 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31235);