Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94193 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1391 |
| commutator | C22×C8 | C42 |
| Frattini, cosocle | #2960 | C22 |
| socle | C2 | #7248 |
| prime radical | C23 | #338 |
| LCS | G ▷ C22×C8 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ #2960 ◁ G | |
| LECS | G ▷ #2960 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,25,12,27,44,7,13,28,42,8,31,32,43)(3,14,16,46,50,47,24,58,18,35,51,61,19,54,55,64)(4,10,21,38,6,29,30,60,22,39,49,41,23,57,26,59)(9,33,34,20,36,56,63,17,37,52,62,15,45,48,40,53);(1,3,6,17,25,50,22,15,7,18,23,53,8,19,4,20)(2,9,10,35,31,45,57,47,13,37,39,14,12,36,29,54)(5,24,26,56,27,51,21,52,28,55,30,48,32,16,49,33)(11,40,41,64,43,62,60,61,42,63,38,58,44,34,59,46)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,d^2=fg,e^2=fgh,g^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gh,h^a=hi,c^b=ci,d^b=dg,d^c=dh,e^d=ei,f^d=fi,g^d=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1463);