Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349137 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
4084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,19)(12,18)(13,16)(14,17)(15,21)(20,23)(22,24);(2,6)(3,4)(7,8)(9,11)(10,15)(12,22)(13,21)(14,20)(16,23)(17,24)(18,19);(10,16)(12,13)(15,23)(17,18)(19,24)(21,22);(10,17)(15,24)(16,18)(19,23);(1,3,5,4)(2,7,8,6)(11,20)(15,24)(19,23)(21,22)Polycyclic presentation:
2^9|e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257406, #72257470 |
| maximal normal subgroup of | #72257406, #72257470 |
| maximal quotient of | #72257948, #72264142 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256380);