Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419459 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1795 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,8)(9,10)(11,17,19,14)(12,15,20,23)(13,16)(18,21)(22,24);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23);(1,3,4,7)(2,6,5,8)(9,12)(10,15,16,23)(11,18)(13,20)(14,21,17,22)(19,24);(2,5)(6,8)(14,17)(21,22)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,c^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225693, #72225695, #72225702, … |
| maximal normal subgroup of | #72225693, #72225695, #72225702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 347973);