Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535696 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,14,19,15,20,13,18)(11,21,22,16,23,24,17,12);(1,2)(3,6)(4,5)(7,8)(9,11,13,17,15,23,14,22)(10,16,19,24,20,12,18,21);(2,5)(6,8)(9,12)(10,17)(11,18)(13,21)(14,24)(15,16)(19,23)(20,22);(1,3)(2,6)(4,7)(5,8)(9,13,15,14)(10,18,20,19)(11,22,23,17)(12,21,16,24);(3,7)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=fi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6452531);