Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #438346 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15617 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1110 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)(3,13)(4,9)(5,21)(6,25)(7,12)(8,16)(10,24)(11,27)(14,39)(15,35)(17,37)(18,45)(19,49)(20,34)(22,29)(23,36)(26,30)(28,50)(31,42)(32,48)(33,40)(38,43)(41,58)(44,51)(46,57)(47,59)(52,54)(53,56)(55,64)(60,62)(61,63);(1,3,6,26,7,17,27,16)(2,8,25,13,12,30,11,37)(4,14,40,42,20,43,19,50)(5,22,23,15,24,53,54,44)(9,31,33,38,34,28,49,39)(10,35,52,56,21,51,36,29)(18,46,61,62,48,64,47,41)(32,58,59,57,45,60,63,55);(1,4,7,20)(2,9,12,34)(3,14)(5,18,24,48)(6,19,27,40)(8,28)(10,32,21,45)(11,33,25,49)(13,38)(15,41)(16,42)(17,43)(22,46)(23,47,54,61)(26,50)(29,55)(30,31)(35,60)(36,59,52,63)(37,39)(44,62)(51,58)(53,64)(56,57);(1,5,6,23,7,24,27,54)(2,10,11,36,12,21,25,52)(3,15,16,22,17,44,26,53)(4,18,19,47,20,48,40,61)(8,29,13,35,30,56,37,51)(9,32,33,59,34,45,49,63)(14,41,42,46,43,62,50,64)(28,55,38,60,31,57,39,58)Polycyclic presentation:
2^9|b^2=eg,c^2=fi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 345543);