Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460664 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16369 |
| 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 | |
893 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)(3,15,19,42)(4,20,25,10)(5,26,30,40)(6,12)(7,14)(8,13)(9,18,34,31)(11,29,41,28)(16,35)(17,33)(21,53)(22,39)(23,38,24,37)(27,52)(32,46)(36,54)(43,47)(44,50)(45,62,56,57)(48,63,64,58)(49,59)(51,60)(55,61);(1,3)(2,9,13,34)(4,21)(5,17)(6,18)(7,31)(8,19)(10,35,20,43)(11,33,41,60)(12,15,14,42)(16,23)(22,48)(24,47)(25,46)(26,44,40,59)(27,56)(28,49)(29,50)(30,51)(32,38,53,37)(36,58,52,63)(39,62,61,57)(45,54)(55,64);(1,4,6,23,8,25,7,24)(2,10,12,37,13,20,14,38)(3,16,18,46,19,47,31,21)(5,27,28,55,30,54,29,22)(9,32,15,43,34,53,42,35)(11,39,40,52,41,61,26,36)(17,48,49,56,51,64,50,45)(33,58,59,62,60,63,44,57);(1,5,8,30)(2,11,13,41)(3,17,19,51)(4,22,25,55)(6,29,7,28)(9,33,34,60)(10,36,20,52)(12,26,14,40)(15,44,42,59)(16,45,47,56)(18,50,31,49)(21,48,46,64)(23,54,24,27)(32,57,53,62)(35,58,43,63)(37,61,38,39)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=e,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,h^b=hi,d^c=de,g^c=gi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367861);