Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C8 | C82 |
| Frattini, cosocle | #1397 | C22 |
| socle | C2 | #7330 |
| prime radical | C22 | #1398 |
| LCS | G ▷ C8 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ G | |
| LECS | G ▷ #1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
59 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,3,9,4,11,17,33,5,13,18,35,21,40,47,59,6,14,19,32,22,41,46,58,23,30,48,55,50,38,62,63,7,12,16,34,20,39,44,60,24,28,45,31,49,56,64,57,25,10,15,36,42,37,43,61,51,8,26,27,52,29,53,54);(2,8,10,28,12,30,14,13)(3,15,16,19)(4,20)(9,31,32,27,34,35,36,55)(11,38,37,40,39,29,41,56)(17,46,44,43)(18,26,45,48)(21,49)(22,42)(33,59,58,63,60,57,61,54)(47,62,64,53)(50,52)Polycyclic presentation:
2^9|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,g^2=h,h^2=i,b^a=bg,d^a=dh,f^a=fi,c^b=chi,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224644, #72224648, #72224652, … |
| maximal normal subgroup of | #72224644, #72224648, #72224652, … |
| maximal quotient of | #72230688, #72234375, C2×#94506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1702);