Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418533 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15263 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,20,34)(4,21,26,38)(5,27,31,11)(6,13,7,12)(9,19,35,18)(10,24,39,25)(16,32,45,48)(17,40,47,56)(22,50,44,36)(23,49,54,62)(28,33,46,59)(29,41,30,42)(37,53,57,55)(43,61,52,58)(51,60,63,64);(1,3)(2,9)(4,22)(5,28)(6,19)(7,18)(8,20)(10,36)(11,40)(12,34)(13,15)(14,35)(16,24)(17,29)(21,48)(23,52)(25,45)(26,44)(27,56)(30,47)(31,46)(32,38)(33,41)(37,61)(39,50)(42,59)(43,54)(49,64)(51,53)(55,63)(57,58)(60,62);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,32)(11,37)(12,38)(13,21)(14,39)(15,36)(17,43)(18,22)(19,44)(20,45)(27,57)(28,51)(29,53)(30,55)(31,54)(33,58)(34,50)(35,48)(40,60)(41,62)(42,49)(46,63)(47,52)(56,64)(59,61);(1,5,6,29,8,31,7,30)(2,11,12,41,14,27,13,42)(3,17,18,28,20,47,19,46)(4,23,24,53,26,54,25,55)(9,33,15,40,35,59,34,56)(10,37,38,62,39,57,21,49)(16,43,22,51,45,52,44,63)(32,58,36,60,48,61,50,64)Polycyclic presentation:
2^9|a^2=i,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,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, 325730);