Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428402 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15383 |
| 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 | |
1020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,26,7,12,29,11)(3,13,27,32,17,38,16,8)(4,18,20,34,21,9,28,39)(5,22,24,37,25,10,35,52)(14,40,45,30,43,57,42,46)(15,36,54,58,44,51,23,31)(19,33,49,62,50,47,55,60)(41,61,53,56,63,59,48,64);(1,3)(2,8)(4,14)(5,23)(6,27)(7,17)(9,30)(10,36)(11,38)(12,32)(13,26)(15,24)(16,29)(18,46)(19,48)(20,45)(21,43)(22,51)(25,54)(28,42)(31,37)(33,59)(34,40)(35,44)(39,57)(41,49)(47,61)(50,53)(52,58)(55,63)(56,60)(62,64);(1,4,7,21)(2,9,12,18)(3,14,17,43)(5,19,25,50)(6,28,29,20)(8,30,32,46)(10,33,22,47)(11,39,26,34)(13,40,38,57)(15,41,44,63)(16,45,27,42)(23,53,54,48)(24,55,35,49)(31,56,58,64)(36,61,51,59)(37,62,52,60);(1,5)(2,10,12,22)(3,15,17,44)(4,19)(6,24)(7,25)(8,31)(9,33,18,47)(11,37,26,52)(13,36)(14,41,43,63)(16,23,27,54)(20,49)(21,50)(28,55)(29,35)(30,56)(32,58)(34,60,39,62)(38,51)(40,61)(42,48,45,53)(46,64)(57,59)Polycyclic presentation:
2^9|a^2=ef,c^2=i,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,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, 335599);