Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15297 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,30)(3,14,17,9,19,34,18,33)(4,10,24,37,23,38,22,41)(5,25,27,11,29,42,28,40)(15,44,20,31,47,35,46,58)(16,48,50,43,49,32,26,39)(21,52,53,36,55,56,54,61)(45,60,51,62,64,57,63,59);(1,3,8,17,7,19,6,18)(2,9,12,34,13,33,30,14)(4,20,24,47,23,46,22,15)(5,26,27,16,29,50,28,49)(10,35,37,58,38,44,41,31)(11,39,42,48,40,43,25,32)(21,45,53,51,55,64,54,63)(36,60,56,62,61,57,52,59);(1,4,7,23)(2,10,13,38)(3,15,19,47)(5,21,29,55)(6,24,8,22)(9,31,33,58)(11,36,40,61)(12,41,30,37)(14,35,34,44)(16,45,49,64)(17,46,18,20)(25,56,42,52)(26,51,50,63)(27,54,28,53)(32,57,48,60)(39,62,43,59);(1,5)(2,11)(3,16)(4,21)(6,28)(7,29)(8,27)(9,32)(10,36)(12,42)(13,40)(14,43)(15,45)(17,50)(18,26)(19,49)(20,51)(22,54)(23,55)(24,53)(25,30)(31,57)(33,48)(34,39)(35,59)(37,56)(38,61)(41,52)(44,62)(46,63)(47,64)(58,60)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328961);