Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #432847 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15467 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,9,29,34,17,35,18,33)(4,19,23,10,24,39,22,38)(5,25,27,11,28,41,30,40)(15,42,48,31,44,36,20,49)(16,45,46,32,26,57,47,61)(21,52,54,50,55,37,56,58)(43,64,53,63,59,60,51,62);(1,3,6,29,8,17,7,18)(2,9,14,33,12,35,13,34)(4,20,22,15,24,48,23,44)(5,26,27,46,28,16,30,47)(10,36,39,31,38,42,19,49)(11,32,25,57,40,61,41,45)(21,53,54,43,55,51,56,59)(37,62,50,64,52,63,58,60);(1,4,8,24)(2,10,12,38)(3,15,17,44)(5,21,28,55)(6,23,7,22)(9,31,35,49)(11,37,40,52)(13,39,14,19)(16,43,26,59)(18,48,29,20)(25,58,41,50)(27,56,30,54)(32,60,61,64)(33,36,34,42)(45,62,57,63)(46,51,47,53);(1,5)(2,11)(3,16)(4,21)(6,30)(7,27)(8,28)(9,32)(10,37)(12,40)(13,25)(14,41)(15,43)(17,26)(18,47)(19,50)(20,51)(22,56)(23,54)(24,55)(29,46)(31,60)(33,45)(34,57)(35,61)(36,62)(38,52)(39,58)(42,63)(44,59)(48,53)(49,64)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340044);