Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6803853 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #39148 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25446 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,14)(4,21)(5,28)(6,11)(7,32)(8,13)(9,18)(10,26)(12,33)(15,45)(16,49)(17,35)(19,38)(20,37)(22,34)(23,39)(24,53)(25,41)(27,40)(29,58)(30,44)(31,42)(36,50)(43,48)(46,61)(47,63)(51,52)(54,57)(55,62)(56,60)(59,64);(1,3,8,20)(2,9)(4,22)(5,16,29,17)(6,30,31,50)(7,18,33,19)(10,34,41,45)(11,35)(12,14)(13,38)(15,27)(21,52,40,43)(23,47)(24,56)(25,51)(26,48)(28,44)(32,37)(36,58)(39,60,62,64)(42,49)(46,55)(53,61,57,63)(54,59);(1,4)(2,10)(3,15)(5,23)(6,24)(7,25)(8,27)(9,34)(11,39)(12,40)(13,41)(14,43)(16,46)(17,47)(18,48)(19,51)(20,22)(21,32)(26,33)(28,57)(29,55)(30,59)(31,54)(35,60)(36,61)(37,52)(38,45)(42,62)(44,63)(49,64)(50,56)(53,58);(1,5,7,6,8,29,33,31)(2,11,12,28,13,42,32,58)(3,16,18,30,20,17,19,50)(4,23,25,24,27,55,26,54)(9,35,14,44,38,49,37,36)(10,39,40,57,41,62,21,53)(15,46,48,59,22,47,51,56)(34,60,43,63,45,64,52,61);(1,6,7,29,8,31,33,5)(2,11,32,58,13,42,12,28)(3,17,19,30,20,16,18,50)(4,24,25,55,27,54,26,23)(9,36,14,35,38,44,37,49)(10,39,21,53,41,62,40,57)(15,47,51,59,22,46,48,56)(34,61,43,60,45,63,52,64)Polycyclic presentation:
2^9|b^2=fgh,d^2=f,e^2=fh,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,f^a=fi,g^a=gi,h^a=hi,c^b=ch,e^b=efh,g^b=gi,h^b=hi,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6711049);