Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #477350 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17109 |
| 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 | |
846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,30,8,12,7,13)(3,14,17,35,18,9,31,36)(4,19,32,39,22,40,23,10)(5,24,27,11,29,43,28,42)(15,45,48,33,20,53,47,37)(16,49,25,34,51,41,52,44)(21,55,58,54,26,38,57,63)(46,61,59,62,50,60,56,64);(1,3)(2,9)(4,20)(5,25)(6,31)(7,17)(8,18)(10,37)(11,41)(12,14)(13,36)(15,22)(16,27)(19,53)(21,56)(23,47)(24,44)(26,59)(28,51)(29,52)(30,35)(32,48)(33,39)(34,43)(38,62)(40,45)(42,49)(46,57)(50,58)(54,60)(55,64)(61,63);(1,4,6,32,8,22,7,23)(2,10,30,19,12,39,13,40)(3,15,17,48,18,20,31,47)(5,26,27,57,29,21,28,58)(9,33,36,53,14,37,35,45)(11,38,43,63,42,55,24,54)(16,50,25,56,51,46,52,59)(34,60,41,64,44,61,49,62);(1,5,8,29)(2,11,12,42)(3,16,18,51)(4,21,22,26)(6,28,7,27)(9,34,14,44)(10,38,39,55)(13,43,30,24)(15,46,20,50)(17,52,31,25)(19,54,40,63)(23,58,32,57)(33,60,37,61)(35,41,36,49)(45,64,53,62)(47,59,48,56)Polycyclic presentation:
2^9|a^2=ef,c^2=ef,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384547);