Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 | |
4655 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)(9,10,15,12)(11,19,22,24)(13,17,14,18)(16,23,20,21);(5,6)(7,8)(9,11,15,22)(10,16,12,20)(13,21,14,23)(17,24,18,19);(2,4)(6,8)(9,12,15,10)(11,20,22,16)(13,17,14,18)(19,21,24,23);(5,6)(7,8)(9,13,15,14)(10,17,12,18)(11,21,22,23)(16,24,20,19);(6,8)(9,13,15,14)(10,18,12,17)(11,21,22,23)(16,19,20,24)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481010);