Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C32 | C23 |
| commutator, socle, prime radical | C22 | C22×C32 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
95 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2)(3,8)(4,6)(5,7)(9,10,12,18,13,19,27,40,14,20,28,41,30,43,53,63,15,11,22,23,31,24,46,48,33,25,47,49,56,50,67,68,16,21,29,42,32,44,54,64,34,45,55,65,57,66,70,72,35,26,17,36,37,51,38,59,58,52,39,60,61,69,62,71);(1,3,5,4)(2,6,7,8);(1,4,5,3)(2,6,7,8)(9,11,12,23,13,24,27,48,14,25,28,49,30,50,53,68,15,21,22,42,31,44,46,64,33,45,47,65,56,66,67,72,16,26,29,36,32,51,54,59,34,52,55,60,57,69,70,71,35,10,17,18,37,19,38,40,58,20,39,41,61,43,62,63)Polycyclic presentation:
2^9|a^2=e,b^2=d,c^2=de,e^2=f,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60796);