Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349215 |
| 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 | |
4182 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)(3,8)(4,6)(5,7)(9,10)(11,19)(12,17)(13,16)(14,18)(15,21)(20,23)(22,24);(2,6)(3,4)(7,8)(9,11)(10,15)(12,22)(13,21)(14,20)(16,23)(17,19)(18,24);(10,16)(12,13)(15,23)(17,18)(19,24)(21,22);(2,7)(6,8);(1,3,5,4)(2,8,7,6)(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=dhi,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257898, #72257915, #72316972 |
| maximal normal subgroup of | #72257898, #72257915, #72316972 |
| maximal quotient of | #72257966, #72264142 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256459);