Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574015 |
| 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 | |
2712 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)(9,10,15,16)(11,13,18,14)(12,17,22,24)(19,23,20,21);(5,6)(7,8)(9,11,14,10,15,18,13,16)(12,20,21,24,22,19,23,17);(2,4)(6,8)(10,16)(11,18)(17,24)(19,20);(5,6)(7,8)(9,12)(10,17)(11,19)(13,23)(14,21)(15,22)(16,24)(18,20);(6,8)(9,13,15,14)(10,18,16,11)(12,21,22,23)(17,19,24,20)Polycyclic presentation:
2^9|a^2=i,b^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481453);