Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60572 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2965 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(3,5)(8,14)(9,13)(10,11)(15,16)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,2,6,9)(3,7,5,13)(4,8,12,16)(10,14,11,15);(1,3,6,5)(2,7,9,13)(4,10,12,11)(8,14,16,15)(17,19,20,26,21,25,27,31)(18,22,23,29,24,28,30,32);(1,4)(2,8)(3,10)(5,11)(6,12)(7,14)(9,16)(13,15)(19,25)(22,28)(26,31)(29,32);(1,3,6,5)(2,7,9,13)(4,11,12,10)(8,15,16,14)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,e^a=eh,f^a=fh,f^b=fh,d^c=di,e^d=eh110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252212);