Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501872 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24913 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33533 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1764 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
643 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)(3,16)(4,23)(5,30)(6,36)(7,13)(8,14)(9,20)(10,27)(11,32)(12,39)(15,17)(18,52)(19,42)(21,53)(22,47)(24,57)(25,43)(26,50)(28,56)(29,49)(31,63)(33,41)(34,64)(35,46)(37,44)(38,55)(40,58)(45,62)(48,61)(51,60)(54,59);(1,3,17,21,7,22,8,40)(2,9,14,44,13,45,15,43)(4,24,28,18,27,60,29,32)(5,31,34,19,35,26,33,61)(6,37,59,20,38,25,39,62)(10,46,56,64,23,30,49,41)(11,50,51,42,52,63,57,48)(12,53,55,16,54,58,36,47);(1,4)(2,10)(3,18)(5,25)(6,31)(7,27)(8,29)(9,41)(11,47)(12,42)(13,23)(14,56)(15,49)(16,52)(17,28)(19,59)(20,33)(21,60)(22,32)(24,40)(26,38)(30,43)(34,62)(35,37)(36,50)(39,61)(44,46)(45,64)(48,54)(51,53)(55,63)(57,58);(1,5)(2,11)(3,19)(4,25)(6,24)(7,35)(8,33)(9,42)(10,47)(12,41)(13,52)(14,51)(15,57)(16,23)(17,34)(18,59)(20,29)(21,26)(22,61)(27,37)(28,62)(30,36)(31,40)(32,39)(38,60)(43,50)(44,63)(45,48)(46,55)(49,58)(53,56)(54,64)Polycyclic presentation:
2^9|b^2=fh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=dh,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=eh,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409068);