Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6355483 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1961 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(6,8)(11,19)(12,13)(15,22)(17,18)(20,21)(23,24);(1,2,3,5,4,6,7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,23)(21,24);(9,11,14,20)(10,15,16,23)(12,19,13,21)(17,22,18,24);(10,16)(15,23)(17,18)(22,24);(2,6)(5,8)(9,12,14,13)(10,17,16,18)(11,19,20,21)(15,22,23,24)Polycyclic presentation:
2^9|b^2=f,c^2=i,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=di,e^b=eh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72339688, #72339696, #72339800, … |
| maximal normal subgroup of | #72339688, #72339696, #72339800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6262720);