Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135531 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8747 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #3487 | C22 |
| LCS | G ▷ C22×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
449 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 449 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,13,17,34)(4,18)(5,22,26,10)(6,11)(7,12)(8,15)(9,20)(14,42)(16,48,50,41)(19,52,24,56)(21,37)(23,31)(25,38,27,36)(28,39)(29,57,59,64)(30,49)(32,53,35,54)(33,55)(40,62)(43,60,46,51)(44,58)(45,63)(47,61);(1,3)(2,8)(4,14)(5,23)(6,16)(7,17)(9,29)(10,34)(11,30)(12,31)(13,22)(15,26)(18,51)(19,46)(20,44)(21,45)(24,43)(25,49)(27,47)(28,50)(32,40)(33,59)(35,58)(36,48)(37,60)(38,41)(39,61)(42,56)(52,63)(53,57)(54,64)(55,62);(1,4,5,24,7,21,26,19)(2,9,10,35,12,33,22,32)(3,14,23,43,17,45,15,46)(6,20,27,53,28,55,25,54)(8,29,34,58,31,59,13,40)(11,37,38,52,39,18,36,56)(16,44,47,57,50,62,49,64)(30,60,41,63,61,51,48,42)Polycyclic presentation:
2^9|a^2=f,c^2=dh,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=di,f^b=fgi,h^b=hi,d^c=di,e^c=egi,f^c=fhi,g^c=gi,h^c=hi,e^d=ei,f^d=fi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42727);