Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1762 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)(11,19)(12,18)(13,16)(14,17)(15,21)(20,23)(22,24);(2,4)(9,11,13,23,14,22,12,21)(10,15,18,24,17,20,16,19);(9,10)(11,20,22,15)(12,16)(13,18)(14,17)(19,21,24,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72234294, #72260879, #72260913, … |
| maximal normal subgroup of | #72234294, #72260879, #72260913, … |
| maximal quotient of | #72234302 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31200);