Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #449972 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15954 |
| 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 | |
969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,9,19,35,17,34,18,33)(4,20,23,37,24,40,30,10)(5,25,28,38,29,39,27,11)(15,41,43,56,21,48,47,31)(16,44,46,57,26,54,45,32)(22,36,51,59,52,58,53,49)(42,55,61,63,50,64,62,60);(1,3,6,19,8,17,7,18)(2,9,14,33,12,34,13,35)(4,21,23,43,24,15,30,47)(5,26,27,45,29,16,28,46)(10,31,40,41,37,56,20,48)(11,32,25,54,38,57,39,44)(22,42,51,62,52,50,53,61)(36,55,49,63,58,64,59,60);(1,4)(2,10,12,37)(3,15,17,21)(5,22)(6,23)(7,30)(8,24)(9,31)(11,36,38,58)(13,20,14,40)(16,42,26,50)(18,43,19,47)(25,49,39,59)(27,51)(28,53)(29,52)(32,55)(33,41)(34,56)(35,48)(44,60)(45,61,46,62)(54,63)(57,64);(1,5,8,29)(2,11,12,38)(3,16,17,26)(4,22,24,52)(6,28,7,27)(9,32,34,57)(10,36,37,58)(13,25,14,39)(15,42,21,50)(18,46,19,45)(20,49,40,59)(23,53,30,51)(31,55,56,64)(33,44,35,54)(41,60,48,63)(43,62,47,61)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357169);