Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500148 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,14,19,15,18,13,17)(11,20,21,24,16,12,22,23);(1,2)(3,6)(4,5)(7,8)(9,11,15,16)(10,12,18,20)(13,21,14,22)(17,23,19,24);(2,5)(3,7)(9,12)(10,16)(11,18)(13,23)(14,24)(15,20)(17,22)(19,21);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,17,18,19)(11,21,16,22)(12,23,20,24);(2,5)(6,8)(9,13,15,14)(10,17,18,19)(11,21,16,22)(12,23,20,24)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6424120);