Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #353 |
| commutator | C42 | C2×C16 |
| Frattini, cosocle | #2150 | C22 |
| socle | C2 | #7063 |
| prime radical | C23 | #349 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #2150 ◁ G | |
| LECS | G ▷ #2150 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,11,6,13,24,36,7,12,23,38,27,39,51,60,8,14,22,37,28,40,50,59,5,10,25,33,26,34,52,57)(3,15,17,41,19,42,46,61,20,31,45,54,48,55,63,64,16,32,43,35,44,56,62,58,18,9,21,29,47,30,49,53);(1,3,5,18,8,16,7,20)(2,9,10,32,14,31,12,15)(4,21,23,17,22,45,25,43)(6,19,26,47,28,44,27,48)(11,35,38,29,37,41,33,54)(13,30,34,56,40,55,39,42)(24,49,51,46,50,63,52,62)(36,58,60,53,59,61,57,64)Polycyclic presentation:
2^9|a^2=d,b^2=eg,c^2=g,d^2=f,e^2=i,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=ci,d^b=degi,d^c=dgi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 925);