Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2760 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14596 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
807 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,28,34)(3,8,16,32,15,31,26,38)(4,17,21,46,20,9,33,36)(5,22,25,39,24,10,27,37)(13,40,42,59,18,29,48,54)(14,30,43,55,23,52,44,56)(19,35,50,58,49,45,51,60)(41,57,61,64,47,53,62,63);(1,3)(2,8,11,31)(4,18)(5,23,24,14)(6,26)(7,15)(9,29,17,40)(10,30)(12,38,34,32)(13,20)(16,28)(19,47,49,41)(21,42)(22,52)(25,43,27,44)(33,48)(35,57)(36,59,46,54)(37,56)(39,55)(45,53)(50,61,51,62)(58,63)(60,64);(1,4,6,21,7,20,28,33)(2,9,34,46,11,17,12,36)(3,13,16,42,15,18,26,48)(5,19,27,51,24,49,25,50)(8,29,38,59,31,40,32,54)(10,35,37,58,22,45,39,60)(14,41,44,62,23,47,43,61)(30,53,55,63,52,57,56,64);(1,5)(2,10)(3,14)(4,19)(6,27)(7,24)(8,30)(9,35)(11,22)(12,39)(13,41)(15,23)(16,44)(17,45)(18,47)(20,49)(21,51)(25,28)(26,43)(29,53)(31,52)(32,56)(33,50)(34,37)(36,60)(38,55)(40,57)(42,62)(46,58)(48,61)(54,64)(59,63)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=fgh,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298197);