Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #59019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,22,26)(19,27,30,24)(20,23,25,21)(28,29,32,31);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,32)(26,27);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,18,23,20,22,26,21,25)(19,28,29,24,30,32,31,27);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20,23,26,22,25,21,18)(19,24,29,32,30,27,31,28);(2,6)(7,8)(10,14)(13,15)(17,21,22,23)(18,25,26,20)(19,29,30,31)(24,32,27,28)Polycyclic presentation:
2^9|a^2=ghi,c^2=fgi,d^2=fgi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7499525);