Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139571 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1910 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #9468 |
| prime radical | #24285 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
504 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 78 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)(5,23,26,9)(6,28)(7,12)(8,15)(10,14)(11,20)(16,46,24,36)(17,50)(18,39)(21,54,25,51)(22,52)(27,33,53,56)(29,44,55,48)(30,59)(31,45)(32,47,34,41)(35,42)(37,40)(38,43)(49,60,57,64)(58,63,61,62);(1,3,14,40,6,17,22,45,7,18,42,59,20,43,4,15)(2,8,19,38,11,30,35,39,12,31,52,50,28,37,10,13)(5,24,41,63,27,57,25,44,26,16,47,62,53,49,21,48)(9,29,51,60,33,58,32,46,23,55,54,64,56,61,34,36);(2,9)(8,29)(10,34)(11,33)(12,23)(13,36)(19,51)(28,56)(30,58)(31,55)(32,35)(37,61)(38,60)(39,46)(50,64)(52,54)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,e^b=ehi,d^c=dh,f^c=fh,e^d=ei,f^e=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72281154, #72342902, #72342931, … |
| maximal normal subgroup of | #72281154, #72342902, #72342931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46767);