Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #414105 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15162 |
| 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 | |
931 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,19,39)(4,20,24,10)(5,25,27,37)(6,13,7,12)(9,29,33,18)(11,26,38,28)(16,43,21,41)(17,47,49,32)(22,52,54,59)(23,36,30,40)(31,45,34,46)(35,53,50,56)(42,55,60,48)(44,58,51,57)(61,64,62,63);(1,3)(2,9)(4,21)(5,17)(6,29)(7,18)(8,19)(10,34)(11,32)(12,39)(13,15)(14,33)(16,24)(20,31)(22,51)(23,45)(25,42)(26,55)(27,49)(28,48)(30,46)(35,58)(36,41)(37,60)(38,47)(40,43)(44,54)(50,57)(52,62)(53,63)(56,64)(59,61);(1,4,6,23,8,24,7,30)(2,10,12,36,14,20,13,40)(3,16,18,45,19,21,29,46)(5,22,26,53,27,54,28,56)(9,31,15,41,33,34,39,43)(11,35,37,59,38,50,25,52)(17,44,48,63,49,51,55,64)(32,57,42,61,47,58,60,62);(1,5,7,28,8,27,6,26)(2,11,13,25,14,38,12,37)(3,17,18,48,19,49,29,55)(4,22,30,56,24,54,23,53)(9,32,15,42,33,47,39,60)(10,35,40,52,20,50,36,59)(16,44,45,63,21,51,46,64)(31,57,41,61,34,58,43,62)Polycyclic presentation:
2^9|a^2=i,c^2=e,d^2=gh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321302);