Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #491760 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×SD16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
497 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,8)(6,7)(9,10)(11,17)(12,15)(13,16)(14,22)(18,21)(19,24)(20,23);(1,2,4,7)(3,6,8,5)(10,15)(16,23)(17,22)(18,24);(10,16)(11,14)(13,21)(15,23)(17,18)(19,20)(22,24);(2,6)(3,8)(5,7)(9,11,12,14)(10,17,15,22)(13,19,20,21)(16,24,23,18)Polycyclic presentation:
2^9|b^2=g,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=dei,e^b=eg,d^c=di,f^c=fh,h^c=hi,e^d=eg50 conjugacy classes — view the full character table.
| maximal subgroup of | #72287259, #72309991, #72320531, … |
| maximal normal subgroup of | #72287259, #72309991, #72320531, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 399021);