Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479233 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2785 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #17193 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
775 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 775 conjugacy classes of subgroups — too many to draw. Only its 152 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 — 4 generators on 64 points:
(1,2,6,12,8,13,7,14)(3,15,18,35,19,36,31,9)(4,20,23,40,24,10,32,39)(5,25,29,44,30,43,28,11)(16,45,21,53,48,37,47,33)(17,34,50,41,51,61,26,58)(22,55,56,52,57,42,27,38)(46,62,54,60,64,59,49,63);(1,3)(2,9)(4,21)(5,26)(6,31)(7,18)(8,19)(10,37)(11,41)(12,36)(13,35)(14,15)(16,23)(17,28)(20,45)(22,46)(24,47)(25,34)(27,54)(29,51)(30,50)(32,48)(33,40)(38,59)(39,53)(42,63)(43,61)(44,58)(49,56)(52,62)(55,60)(57,64);(1,4)(2,10)(3,16)(5,27)(6,32)(7,23)(8,24)(9,33)(11,42)(12,40)(13,20)(14,39)(15,37)(17,49)(18,47)(19,48)(21,31)(22,28)(25,52)(26,46)(29,57)(30,56)(34,60)(35,53)(36,45)(38,43)(41,62)(44,55)(50,64)(51,54)(58,59)(61,63);(1,5)(2,11,13,44)(3,17)(4,22)(6,29)(7,28)(8,30)(9,34,35,61)(10,38,20,52)(12,25,14,43)(15,41,36,58)(16,46)(18,50)(19,51)(21,54)(23,56)(24,57)(26,31)(27,32)(33,59,53,62)(37,60,45,63)(39,55,40,42)(47,49)(48,64)Polycyclic presentation:
2^9|a^2=fg,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386429);