Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471189 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16773 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,18,9,20,41,19,34)(4,21,24,37,26,36,25,10)(5,27,29,40,31,39,30,11)(16,32,46,52,22,54,45,42)(17,47,51,62,50,33,49,43)(23,56,57,53,58,35,28,38)(44,60,48,59,55,61,63,64);(1,3,6,19,8,20,7,18)(2,9,13,15,14,34,12,41)(4,22,24,45,26,16,25,46)(5,17,29,51,31,50,30,49)(10,32,21,52,37,54,36,42)(11,33,27,43,40,47,39,62)(23,55,28,48,58,44,57,63)(35,59,53,60,56,64,38,61);(1,4)(2,10)(3,16,20,22)(5,28)(6,25)(7,24)(8,26)(9,32,34,54)(11,38)(12,21)(13,36)(14,37)(15,42,41,52)(17,48,50,63)(18,46,19,45)(23,29)(27,56)(30,58)(31,57)(33,60,47,61)(35,39)(40,53)(43,59,62,64)(44,51,55,49);(1,5,6,29,8,31,7,30)(2,11,12,39,14,40,13,27)(3,17,19,51,20,50,18,49)(4,23,24,28,26,58,25,57)(9,33,41,62,34,47,15,43)(10,35,36,38,37,56,21,53)(16,44,46,63,22,55,45,48)(32,59,42,61,54,64,52,60)Polycyclic presentation:
2^9|a^2=eg,b^2=h,c^2=ef,d^2=fi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378386);