Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17090 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,6,12)(3,15,19,9,20,34,18,35)(4,21,24,37,25,41,31,10)(5,26,29,39,30,11,28,40)(16,44,46,52,22,32,51,42)(17,43,49,57,50,61,48,33)(23,53,27,36,55,38,56,63)(45,59,47,60,54,62,58,64);(1,3)(2,9)(4,22)(5,17)(6,19)(7,18)(8,20)(10,32)(11,33)(12,34)(13,15)(14,35)(16,25)(21,52)(23,45)(24,46)(26,57)(27,58)(28,49)(29,48)(30,50)(31,51)(36,62)(37,44)(38,60)(39,43)(40,61)(41,42)(47,56)(53,64)(54,55)(59,63);(1,4,6,24,8,25,7,31)(2,10,12,21,14,37,13,41)(3,16,18,46,20,22,19,51)(5,27,28,55,30,56,29,23)(9,32,15,42,35,44,34,52)(11,38,39,63,26,53,40,36)(17,47,48,54,50,58,49,45)(33,60,61,62,57,64,43,59);(1,5,8,30)(2,11,14,26)(3,17,20,50)(4,23,25,55)(6,29,7,28)(9,33,35,57)(10,36,37,63)(12,40,13,39)(15,43,34,61)(16,45,22,54)(18,49,19,48)(21,53,41,38)(24,56,31,27)(32,59,44,62)(42,64,52,60)(46,58,51,47)Polycyclic presentation:
2^9|a^2=f,c^2=e,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383681);