Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510223 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31396 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #28183 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,15,9,12,6,13)(3,10,18,29,17,5,26,27)(4,20,24,41,23,47,37,11)(7,31,35,42,36,14,30,46)(16,48,50,38,51,57,49,22)(19,28,32,62,45,34,61,43)(21,54,52,58,56,53,55,39)(25,60,33,40,59,44,63,64);(1,3)(2,10,12,5)(4,21)(6,18)(7,32)(8,26)(9,17)(11,39,41,58)(13,29,15,27)(14,43,31,62)(16,25)(19,35)(20,53,47,54)(22,40,38,64)(23,56)(24,55)(28,46,34,42)(30,45)(33,49)(36,61)(37,52)(44,48,60,57)(50,63)(51,59);(1,4,8,37,9,23,6,24)(2,11,13,20,12,41,15,47)(3,16,26,50,17,51,18,49)(5,22,27,57,10,38,29,48)(7,33,35,25,36,63,30,59)(14,44,42,64,31,60,46,40)(19,52,61,56,45,55,32,21)(28,53,62,58,34,54,43,39);(1,5,9,10)(2,3,12,17)(4,22,23,38)(6,29,8,27)(7,34,36,28)(11,16,41,51)(13,26,15,18)(14,45,31,19)(20,50,47,49)(21,40,56,64)(24,48,37,57)(25,39,59,58)(30,62,35,43)(32,46,61,42)(33,54,63,53)(44,55,60,52)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=eh,d^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,e^c=ei,g^c=gh,h^c=hi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417420);