Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500819 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24260 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #33572 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
997 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)(3,17)(4,22)(5,10)(6,14)(7,13)(8,32)(9,16)(11,25)(12,21)(15,35)(18,51)(19,38)(20,43)(23,59)(24,49)(26,58)(27,45)(28,47)(29,44)(30,42)(31,46)(33,50)(34,63)(36,53)(37,55)(39,56)(40,54)(41,52)(48,62)(57,60)(61,64);(1,3,6,5,9,21,7,19)(2,10,13,17,16,38,14,12)(4,23,25,60,27,56,30,61)(8,33,29,55,35,20,31,28)(11,39,22,57,42,59,45,64)(15,47,44,43,32,37,46,50)(18,52,24,34,53,62,54,26)(36,58,49,48,51,63,40,41);(1,4)(2,11)(3,18)(5,24)(6,25)(7,30)(8,34)(9,27)(10,36)(12,40)(13,22)(14,45)(15,48)(16,42)(17,49)(19,54)(20,56)(21,53)(23,33)(26,35)(28,61)(29,62)(31,52)(32,41)(37,59)(38,51)(39,47)(43,57)(44,63)(46,58)(50,64)(55,60);(1,5,6,21,9,19,7,3)(2,12,13,10,16,17,14,38)(4,24,25,53,27,54,30,18)(8,28,29,33,35,55,31,20)(11,40,22,36,42,49,45,51)(15,43,44,37,32,50,46,47)(23,26,60,52,56,34,61,62)(39,48,57,63,59,41,64,58)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408015);