Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31282 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2684 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14365 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,14)(3,9,18,34,17,35,30,33)(4,19,22,10,23,39,29,38)(5,24,27,11,28,41,26,40)(15,42,45,60,44,36,20,31)(16,46,47,32,25,56,48,61)(21,51,53,57,55,37,54,49)(43,59,50,62,58,64,52,63);(1,3,6,18,8,17,7,30)(2,9,13,33,12,35,14,34)(4,20,22,44,23,45,29,15)(5,25,26,48,28,16,27,47)(10,36,19,31,38,42,39,60)(11,32,41,46,40,61,24,56)(21,52,53,58,55,50,54,43)(37,62,57,64,51,63,49,59);(1,4)(2,10)(3,15)(5,21)(6,29)(7,22)(8,23)(9,31)(11,37)(12,38)(13,39)(14,19)(16,43)(17,44)(18,45)(20,30)(24,57)(25,58)(26,54)(27,53)(28,55)(32,59)(33,36)(34,42)(35,60)(40,51)(41,49)(46,63)(47,50)(48,52)(56,62)(61,64);(1,5)(2,11)(3,16)(4,21)(6,27)(7,26)(8,28)(9,32)(10,37)(12,40)(13,24)(14,41)(15,43)(17,25)(18,48)(19,49)(20,50)(22,54)(23,55)(29,53)(30,47)(31,59)(33,56)(34,46)(35,61)(36,62)(38,51)(39,57)(42,63)(44,58)(45,52)(60,64)Polycyclic presentation:
2^9|a^2=h,b^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,g^c=gi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291082);