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 | #8609 |
| 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)(4,19)(5,23,24,9)(6,26)(7,12)(8,15)(10,14)(11,20)(16,43,44,33)(17,46)(18,36)(21,50,51,47)(22,48)(25,31,49,54)(27,41,52,62)(28,56)(29,42)(30,38,53,60)(32,39)(34,37)(35,40)(45,58,61,63)(55,64,57,59);(1,3,14,37,6,17,22,42,7,18,39,56,20,40,4,15)(2,8,19,35,11,28,32,36,12,29,48,46,26,34,10,13)(5,16,38,59,25,45,51,62,24,44,60,64,49,61,21,41)(9,27,47,58,31,55,53,43,23,52,50,63,54,57,30,33);(2,9)(8,27)(10,30)(11,31)(12,23)(13,33)(19,47)(26,54)(28,55)(29,52)(32,53)(34,57)(35,58)(36,43)(46,63)(48,50)Polycyclic presentation:
2^9|a^2=f,b^2=dg,d^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,f^c=fh,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72287260, #72287275, #72287291, … |
| maximal normal subgroup of | #72287260, #72287275, #72287291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41969);