Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7333 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #322 |
| commutator | C8 | C82 |
| Frattini, cosocle | #1726 | C22 |
| socle | C2 | #7014 |
| prime radical | C23 | H(3,ℤ/4) |
| LCS | G ▷ C8 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ #1726 ◁ G | |
| LECS | G ▷ #1726 ▷ C22×C4 ▷ C2 ▷ C1 | |
191 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 191 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,10)(3,14,16,41,18,44,48,63)(4,19,20,50)(6,12,25,37)(7,13,24,38)(8,27,28,55,31,56,58,46)(9,32,33,22)(11,39,40,26)(15,29,45,59,47,30,57,36)(17,43,23,53,49,42,54,64)(21,34,51,62)(35,60,61,52);(1,3)(2,8,11,29,13,31,12,30)(4,15)(5,23,24,54)(6,17)(7,18)(9,14,34,42,19,44,35,43)(10,36,37,58,38,59,40,28)(16,25,48,39)(20,27,32,56)(21,46)(22,47)(26,49)(33,53,61,63,50,64,62,41)(45,51,57,60)(52,55)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,b^a=bc,c^a=ci,e^a=ef,h^a=hi,d^b=df,g^b=gi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72283823, #72283870, #72283893, … |
| maximal normal subgroup of | #72283823, #72283870, #72283893, … |
| maximal quotient of | C2×#93253 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 449);