Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,14,19,16,20,15,13)(11,21,24,17,23,12,18,22);(5,6)(7,8)(9,11)(10,17)(12,13)(14,18)(15,24)(16,23)(19,21)(20,22);(1,2,3,4)(6,8)(9,12,15,17,16,21,14,22)(10,18,13,23,20,24,19,11);(6,8)(9,13,15,20,16,19,14,10)(11,22,18,12,23,17,24,21);(9,14,16,15)(10,19,20,13)(11,18,23,24)(12,17,21,22)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,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351565);