Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| 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 | C2×#3061 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2648 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)(9,10,13,16,14,19,12,18)(11,17,21,23,22,24,20,15);(5,6)(7,8)(9,11,14,22)(10,15,19,23)(12,21,13,20)(16,24,18,17);(1,2,3,4)(6,8)(9,12,14,13)(10,16,19,18)(11,20,22,21)(15,17,23,24);(6,8)(9,11)(10,17)(12,20)(13,21)(14,22)(15,18)(16,23)(19,24);(9,12,14,13)(10,18,19,16)(11,21,22,20)(15,17,23,24)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350262);