Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152123 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12697 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | #3499 | C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #117 ◁ #3509 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
619 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 619 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,5,11,7,13,26,41)(3,14,17,39,16,33,45,59)(4,19,22,48,21,9,34,36)(6,12,28,40,29,10,25,38)(8,24,32,27,31,46,58,18)(15,42,43,61,20,49,50,64)(23,35,55,60,52,37,54,62)(30,56,57,51,47,53,63,44);(1,3)(2,8)(4,20)(5,24)(6,27)(7,16)(9,30)(10,33)(11,39)(12,14)(13,31)(15,21)(17,28)(18,29)(19,47)(22,53)(23,44)(25,45)(26,46)(32,38)(34,56)(35,49)(36,61)(37,42)(40,58)(41,59)(43,54)(48,64)(50,55)(51,52)(57,60)(62,63);(1,4)(2,9,13,19)(3,15)(5,22)(6,23)(7,21)(8,30,31,47)(10,35,12,37)(11,36,41,48)(14,42,33,49)(16,20)(17,43)(18,44)(24,56)(25,54)(26,34)(27,51)(28,55)(29,52)(32,57,58,63)(38,60,40,62)(39,61,59,64)(45,50)(46,53)Polycyclic presentation:
2^9|a^2=f,c^2=hi,d^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dhi,e^b=ei,f^b=fgh,g^b=gi,e^c=ei,f^d=fhi,g^f=gi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59322);