Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #449603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15948 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1929 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,29,6,12,8,13)(3,14,30,35,17,9,18,34)(4,19,22,38,23,10,31,39)(5,24,27,11,26,42,28,41)(15,32,45,36,46,43,20,50)(16,47,49,40,25,58,48,33)(21,53,54,37,55,57,56,51)(44,63,52,62,59,61,64,60);(1,3)(2,9)(4,20)(5,25)(6,17)(7,30)(8,18)(10,36)(11,40)(12,14)(13,35)(15,22)(16,26)(19,50)(21,52)(23,45)(24,47)(27,48)(28,49)(29,34)(31,46)(32,38)(33,41)(37,61)(39,43)(42,58)(44,56)(51,63)(53,62)(54,59)(55,64)(57,60);(1,4)(2,10)(3,15)(5,21)(6,23)(7,31)(8,22)(9,32)(11,37)(12,19)(13,39)(14,43)(16,44)(17,46)(18,45)(20,30)(24,57)(25,59)(26,55)(27,56)(28,54)(29,38)(33,60)(34,50)(35,36)(40,62)(41,51)(42,53)(47,61)(48,52)(49,64)(58,63);(1,5,6,26)(2,11,12,41)(3,16,17,25)(4,21,23,55)(7,28,8,27)(9,33,14,40)(10,37,19,51)(13,42,29,24)(15,44,46,59)(18,49,30,48)(20,52,45,64)(22,56,31,54)(32,60,43,62)(34,58,35,47)(36,61,50,63)(38,57,39,53)Polycyclic presentation:
2^9|a^2=eh,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,g^c=gi,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, 356800);