Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #449385 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15941 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
811 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)(3,15)(4,21)(5,26)(6,14)(7,13)(8,12)(9,18)(10,25)(11,29)(16,43)(17,47)(19,42)(20,34)(22,32)(23,52)(24,37)(27,39)(28,41)(30,40)(31,38)(33,49)(35,46)(36,56)(44,60)(45,58)(48,62)(50,53)(51,63)(54,61)(55,59)(57,64);(1,3,8,20,7,19,6,18)(2,9,14,42,13,34,12,15)(4,22,31,46,25,45,24,16)(5,27,28,49,30,48,29,17)(10,35,38,32,21,43,37,58)(11,39,40,47,41,62,26,33)(23,44,53,55,54,64,56,51)(36,60,61,63,50,57,52,59);(1,4,8,31,7,25,6,24)(2,10,12,37,13,21,14,38)(3,16,18,45,19,46,20,22)(5,23,29,56,30,54,28,53)(9,32,42,43,34,58,15,35)(11,36,40,61,41,50,26,52)(17,44,27,55,49,64,48,51)(33,57,62,63,47,60,39,59);(1,5,7,30)(2,11)(3,17)(4,23,25,54)(6,28,8,29)(9,33,34,47)(10,36)(12,40)(13,41)(14,26)(15,39,42,62)(16,44)(18,27)(19,49)(20,48)(21,50)(22,51)(24,53,31,56)(32,57,58,60)(35,59,43,63)(37,61)(38,52)(45,55)(46,64)Polycyclic presentation:
2^9|b^2=ef,c^2=gi,d^2=eg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356582);