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 | |
3273 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10,16,19)(11,20,23,24)(12,15,21,14)(13,18,17,22);(5,6)(7,8)(9,11,15,18,16,23,14,22)(10,17,12,20,19,13,21,24);(2,4)(6,8)(9,12,15,19,16,21,14,10)(11,20,18,13,23,24,22,17);(6,8)(9,13)(10,18)(11,21)(12,23)(14,20)(15,24)(16,17)(19,22);(9,14,16,15)(10,12,19,21)(11,22,23,18)(13,24,17,20)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346192);