Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #504058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #26300 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,14)(5,15)(6,12)(8,11)(9,13)(10,18)(16,17)(19,28)(20,23)(21,30)(22,29)(24,27)(25,26)(31,32);(1,3,7,2,9,12,6,13)(4,16,18,32,19,26,22,27)(5,15,20,11,21,30,8,23)(10,25,14,31,29,17,28,24);(1,4,7,22,9,19,6,18)(2,10,3,14,13,29,12,28)(5,17,20,31,21,25,8,24)(11,26,15,27,23,16,30,32);(1,5,6,8,9,21,7,20)(2,11,3,15,13,23,12,30)(4,17,18,24,19,25,22,31)(10,26,14,27,29,16,28,32)Polycyclic presentation:
2^9|b^2=f,c^2=efh,d^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411254);