Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134747 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1812 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #8608 |
| prime radical | #525 | D4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ #2113 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
302 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 84 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 — 3 generators on 64 points:
(1,2)(3,13,16,34)(4,18)(5,22,23,9)(6,25)(7,12)(8,14,28,37)(10,20)(11,26)(15,39)(17,43,42,35)(19,46,47,44)(21,45)(24,31,53,52)(27,36)(29,56,54,38)(30,48,51,63)(32,49)(33,41)(40,57)(50,59)(55,64)(58,60)(61,62);(1,3,4,14,6,17,20,38,7,16,21,37,26,42,49,56)(2,8,10,13,11,29,18,35,12,28,32,34,25,54,45,43)(5,15,19,36,24,40,48,58,23,41,47,59,53,62,63,64)(9,27,30,33,31,55,44,57,22,50,51,39,52,60,46,61);(2,9)(8,27)(10,30)(11,31)(12,22)(13,33)(18,44)(25,52)(28,50)(29,55)(32,51)(34,39)(35,57)(43,61)(45,46)(54,60)Polycyclic presentation:
2^9|a^2=f,b^2=d,d^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,f^b=fi,f^c=fh,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72287291, #72342989, #72343002 |
| maximal normal subgroup of | #72287291, #72342989, #72343002 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41943);