Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3497 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
10740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(6,8)(9,10,16,20)(11,21,13,17)(12,23,14,18)(15,22,24,19);(5,6,7,8)(9,11,16,13)(10,17,20,21)(12,22,14,19)(15,18,24,23);(9,12,16,14)(10,18,20,23)(11,22,13,19)(15,17,24,21);(9,13,16,11)(10,17,20,21)(12,22,14,19)(15,23,24,18);(1,3,4,2)(9,14,16,12)(10,18,20,23)(11,22,13,19)(15,21,24,17);(9,15)(10,19)(11,18)(12,17)(13,23)(14,21)(16,24)(20,22)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,d^2=h,e^2=i,b^a=bg,c^a=ch,f^a=fh,e^b=eh,d^c=dh164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532944);