Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35273 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,14,21,16,19,15,18)(11,20,24,12,22,13,17,23);(5,6)(7,8)(9,11)(10,12)(13,18)(14,17)(15,24)(16,22)(19,23)(20,21);(1,2,3,4)(6,8)(9,12,14,13,16,23,15,20)(10,11,21,24,19,22,18,17);(6,8)(9,13,15,12,16,20,14,23)(10,17,18,22,19,24,21,11);(9,14,16,15)(10,18,19,21)(11,17,22,24)(12,13,23,20)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351553);